View source for 3-ary Boolean functions

{{Boolf header}}
[[File:Venn 0101 0011.svg|thumb|The 3-ary Boolean function <math>\scriptstyle ( A \land \neg C ) \lor ( B \land C )</math> represented as a Venn diagram ]]
[[File:Boolean functions like 1100 1010.svg|thumb|''sec'' 53 containing 8 functions, including the one shown above (shown in the top right position)]]
[[File:3-ary Boolean functions; super-clan of Zhe 44.svg|thumb|''bec'' 27 containing 3 ''sec''s, including the one shown above (shown in orange)]]
[[File:Venn and Euler diagrams of 3-ary Boolean relations; with graphs.svg|thumb|right|The 22 ''bec''s ordered by weight, including the one shown above (shown second on the left in box 4)<br>Compare {{oeis|A039754}}]]

There are <math>2^{2^{3}}</math> = 256 [[w:Ternary operation|3-ary]] [[w:Boolean function|Boolean functions]], like [[w:Set (mathematics)|set operations]] or [[w:Logical connective|logical connectives]].

This [[w:Venn diagram|Venn diagram]] [[File:Venn 0000 0001.svg|35px|0000 0001]], representing the [[w:intersection (set theory)|intersection]] of 3 sets, or the [[w:logical conjunction|conjunction]] of 3 statements respectively, gives an example of a 3-ary Boolean function. 

__TOC__


==Equivalence classes==

::''See: [[Equivalence classes of Boolean functions]]''

The set of 256 functions can be [[w:Partition of a set|parted]] into various [[w:Equivalence class|equivalence classes]].<br>
The most basic way to do that are the 46 [[Equivalence classes of Boolean functions#sec|small equivalence classes]] (''sec''s). Functions belong to the same ''sec'', when they can be expressed by each other by negating arguments. There are 3 arguments that can be negated, so there can be up to  <math>2^{3}</math> = 8 different functions in a ''sec''.

When binarily colored cubes can be transformed into each other by mirroring and rotation, they are essentially the same.<br>
The corresponding Boolean functions are often called ''equivalent'' and belong to the same [[Equivalence classes of Boolean functions#bec|big equivalence class]] (''bec'').<br>


{| class="wikitable" style="text-align: center; width: 800px;"
|+ Numbers of equivalence classes
| colspan="2"| ''wec'': '''5''' 
|-
| ''sec'': {{oeis|A000231}}(3) = '''46'''
| ''bec'': {{oeis|A000616}}(3) = '''22'''
|-
| ''gsec'': '''30'''
| ''gbec'': '''14'''
|-
| ''ggsec'': '''18'''
| ''ggbec'': '''8'''
|}


The following sequences show the number of ''sec''s and ''bec''s by weight:
                 <span style="color: gray;">weight   =   0   1   2   3   4   5   6   7   8</span>
 ''sec''s: [[oeis:A054724|A054724]](3, 0..8)   =   1   1   7   7  14   7   7   1   1
 ''bec''s: [[oeis:A039754|A039754]](3, 0..8)   =   1   1   3   3   6   3   3   1   1

The following sequences show the number of ''sec''s and ''bec''s by number of functions in each ''sec'':
                              <span style="color: gray;">1    2    4    8</span>
 ''sec''s: [[oeis:A227725|A227725]](3, 0..3)   =   2    7   14   23
 ''bec''s:                        2    3    6   11


===Permutations of functions in a ''bec''===

Theoretically a ''bec'' could have 3! * 2<sup>3</sup> = 6 * 8 = 48 functions, but the largest ''bec''s have only 24. <small>(There are some ''bec''s of [[4-ary Boolean functions|4-ary functions]] with the highest possible number of functions, e.g. [[:File:1111 1100 1001 0000 Boolean function 16*24.svg#File|this one]].)</small>

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#bbb"| permuting all 3 arguments
|-
|
{| style="border-spacing: 0; margin: 1em auto; text-align: center;"
|style="padding:20px;"| [[File:3-ary Boolean functions; cube permutations; 3.svg|thumb|center|600px|3]]
|style="padding:20px;"| [[File:3-ary Boolean functions; cube permutations; 4.svg|thumb|center|600px|4]]
|}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#bbb" colspan="3"| permutations of functions in a ''sec''
|-
|style="width:300px; vertical-align:top;"| [[File:Subgroup of Oh; S3 green 03; cycle graph.svg|thumb|280px|Cycle graph of [[w:Symmetric group|S]]<sub>3</sub><br>Each of the six files corresponds to a permutation of (A, B, C).]]
|[[File:3-ary Boolean functions; cube permutations; 0.svg|thumb|center|600px|0]]
|style="width:300px; vertical-align:top;"| [[File:Full octahedral group; cycle graph.svg|thumb|280px|All 48 permutations in the cycle graph of the [[full octahedral group]]]]
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa;"
! bgcolor="#bbb"| permuting 2 arguments
|-
|
{| style="border-spacing: 0; margin: 1em auto; text-align: center;"
|style="padding:20px;"| [[File:3-ary Boolean functions; cube permutations; 1.svg|thumb|center|400px|1]]
|style="padding:20px;"| [[File:3-ary Boolean functions; cube permutations; 5.svg|thumb|center|400px|5]]
|style="padding:20px;"| [[File:3-ary Boolean functions; cube permutations; 2.svg|thumb|center|400px|2]]
|}
|}

==Nonlinearity==

The nonlinearity of a function is the smallest [[w:Hamming distance|Hamming distance]] it has from one of the 16 linear functions, i.e. from a row of the binary [[w:Walsh matrix|Walsh matrix]] or its complement.<br>
The functions in ''wec'' E0 are the linear functions, i.e. those with nonlinearity 0. The functions in ''wec'' O have nonlinearity 1, and all others have nonlinearity 2.<br>
There are no 3-ary [[w:Bent function|bent functions]].

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" colspan="3"|Nonlinearity
|-
| [[File:Binary Walsh matrix of order 8 with row numbers.svg|thumb|center|250px|The 8 functions in the Walsh matrix and their complements are linear.]]
| [[File:0111 0000 nonlinearity.svg|thumb|center|200px|<math>\scriptstyle ( A \lor B ) \land \neg C</math> in ''sec'' 7 has nonlinearity 1.]]
| [[File:1100 1010 nonlinearity.svg|thumb|center|200px|<math>\scriptstyle ( \neg A \lor \neg C ) \land (\neg B \lor C)</math> in ''sec'' 53 has nonlinearity 2.]]
|}

==Walsh spectrum==


[[File:1010 0110 Walsh spectrum (single row).svg|thumb|right|300px|The Boolean function <math>\scriptstyle (1,0,1,0,0,1,1,0)</math><br>has the Walsh spectrum <math>\scriptstyle (4,2,0,-2,0,2,0,2)</math><ref>Compare Figure 1 in [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.74.8029&rep=rep1&type=pdf Walsh Spectrum Computations Using Cayley Graphs], by W. J. Townsend and M. A. Thornton</ref><br>and the binary Walsh spectrum <math>\scriptstyle (0,1,0,1,0,1,0,1)</math>. The latter is row 1 of the Walsh matrix.]]

The [[w:Hadamard transform|Walsh spectrum]] of a Boolean function is the [[w:Matrix multiplication|product]] of its binary string (as a row vector) with a [[w:Walsh matrix|Walsh matrix]].<br>
The first entry of the Walsh spectrum is the functions digit sum, and all entries have the same [[w:Parity (mathematics)|parity]].

In the following chapter Walsh spectra are shown in the "by ''ggbec''" sections.<br>
Walsh spectra of the same ''sec'' (which differ only in the [[w:Sign (mathematics)|signs]] of their entries) are always shown in the same file.

The Walsh spectra of complements sum up to <math>\scriptstyle (8,0,0,0,0,0,0,0,)</math>.
<ref>Compare Property 1 (page 3 of the PDF) in [http://matwbn.icm.edu.pl/ksiazki/amc/amc13/amc13413.pdf The spectral test of the Boolean function linearity] by P. Porwik</ref>


[[File:Binary Walsh matrix of order 8 with row numbers.svg|thumb|left|150px|Walsh matrix with row numbers]]

Further the [[Equivalence classes of Boolean functions#Binary Walsh spectrum|'''binary Walsh spectra''']] are always shown by the red squares in the background. They are always rows of a binary Walsh matrix - or, in other terms, [[w:exclusive or|exclusive disjunctions]] of unnegated arguments. 

The functions with odd digit sum, which make up [[#O|''wec'' O]], have each binary Walsh spectrum one time in every ''sec''.

The functions with even digit sum, which make up the other four ''wec''s, have always the same binary Walsh spectrum in the same ''sec'':

{|
|-
|<math>\scriptstyle \oplus()</math> ||= 0000 0000|| || || || || || || || || || || ||in [[#E0|''wec'' E0]]
|-
|<math>\scriptstyle \oplus(A)</math> ||= 0101 0101||&nbsp;&nbsp;&nbsp;||or||&nbsp;&nbsp;&nbsp;||<math>\scriptstyle \oplus(B)</math> ||= 0011 0011||&nbsp;&nbsp;&nbsp;||or||&nbsp;&nbsp;&nbsp;||<math>\scriptstyle \oplus(C)</math> ||= 0000 1111||&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;||in [[#E1|''wec'' E1]]
|-
|<math>\scriptstyle \oplus(A,B)</math> ||= 0110 0110||&nbsp;&nbsp;&nbsp;||or||&nbsp;&nbsp;&nbsp;||<math>\scriptstyle \oplus(A,C)</math> ||= 0101 1010||&nbsp;&nbsp;&nbsp;||or||&nbsp;&nbsp;&nbsp;||<math>\scriptstyle \oplus(B,C)</math> ||= 0011 1100||&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;||in [[#E2|''wec'' E2]]
|-
|<math>\scriptstyle \oplus(A,B,C)</math> ||= 0110 1001|| || || || || || || || || || || ||in [[#E3|''wec'' E3]]
|}


===Functions with the same binary Walsh spectrum===


In the diagrams in this section the even functions are shown in red, and the odd ones <small>(i.e. those in ''wec'' O)</small> in yellow. Each odd function differs in the lowest bit <small>(numerically: by 1)</small> from an even function with the same binary Walsh spectrum. So in the octeract graphs each red vertex is connected to a yellow vertex by an edge of the direction corresponding to the lowest bit <small>(like the edge connecting vertices 0 and 1)</small>. The pattern of the red vertices is always symmetrical to the main axes. Surprisingly the yellow vertices are also symmetrical, but to axes tilted to the left or to the right (indicated by \ or / in the table).

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#ffb4b4"| 3
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|\<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 7.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 7.svg|87px]]<br>7
|}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#666"| 2
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|\<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 3.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 3.svg|87px]]<br>3
|/<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 5.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 5.svg|87px]]<br>5
|\<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 6.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 6.svg|87px]]<br>6
|}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#bbb"| 1
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|/<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 1.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 1.svg|87px]]<br>1
|\<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 2.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 2.svg|87px]]<br>2
|/<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 4.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 4.svg|87px]]<br>4
|}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa;"
! bgcolor="#e60000"| 0
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|/<br>[[File:3-ary Boolean functions in octeract graph; binary Walsh spectrum 0.svg|250px]]<br>[[File:3-ary Boolean functions; binary Walsh spectrum 0.svg|87px]]<br>0
|}
|}

==Nested equivalence classes==


===Overview===


[[File:3-ary Boolean functions; tribes.svg|right|600px]]

The matrix on the right shows the 256 functions arranged in a way similar to the [[w:Hasse diagram|Hasse diagram]], which is an [[:File:Octeract Petrie polygon.svg|octeract graph]].<br>
The colors indicate the five ''[[Equivalence classes of Boolean functions#wec|wec]]''s, as shown in the table below.<br>

The two following matrices are examples of a ''gbec'' and a ''ggsec''. Both contain the ''sec'' 26. Files like these are in the collapsible boxes below in boxes labeled ''Positions''.

These matrices are symmetric to the main diagonal, because they contain complete ''sec''s. They are also symmetric to the antidiagonal, because they contain all the complements, and complements are symmetric about the center of the matrix. The ''ggsec'' matrices are also symmetric about the central axes, because they contain all the half-complements. Left half-complements are vertically, and right ones are horizontally symmetric to each other.

{| align="left"
| [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|thumb|300px|left|''gbec'' 25 with 6 ''sec''s (''sec'' 26 is light green)]]
| [[File:3-ary Boolean functions; ggsec 21.svg|thumb|300px|left|''ggsec'' 21 with 4 ''sec''s (''sec'' 26 is light red)]]
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" colspan="2"|Hasse diagram and diamond matrix
|-
|style="vertical-align:top;"| [[File:Octeract Petrie polygon.svg|700px|thumb|center|The [[w:Hasse diagram|Hasse diagram]] of the 256 functions ordered by implication is an [[w:8-cube|8-dimensional cube]] or ''octeract'', here represented in its [[w:Petrie polygon|Petrie polygon]] projection.]]
|style="vertical-align:top;"| [[File:16x16_matrix_like_octeract_Hasse_diagram.svg|700px|thumb|center|The same matrix as above in diamond orientation — Instead of decimal numbers it shows the [[User:Watchduck/hat#Nibble_shorthands|nibble shorthands]] that are also used in the ''sec'' files. ]]
|-
|colspan="2"| On the bottom of each ''sec'' file (e.g. [https://upload.wikimedia.org/wikipedia/commons/b/b8/Boolean_functions_like_1000_0000.svg this] one) the positions of the functions are shown in the octeract graph and the diamond matrix.
|}


{| class="wikitable" style="text-align:center; background:white; width:100%;"
|-
|style="width:20%"|'''<big>[[#wec O with 128 functions|O]]</big>'''<br><small>{{font|color=gray|nonlinearity 1}}</small>
|style="width:20%"|'''<big><span style="background:#e60000">&nbsp; [[#wec E0 with 16 functions|E0]] &nbsp;</span></big>'''<br><small>{{font|color=gray|nonlinearity 0}}</small>
|style="width:20%"|'''<big><span style="background:#bbbbbb">&nbsp; [[#wec E1 with 48 functions|E1]] &nbsp;</span></big>'''<br><small>{{font|color=gray|nonlinearity 2}}</small>
|style="width:20%"|'''<big><span style="background:#666666">&nbsp; [[#wec E2 with 48 functions|E2]] &nbsp;</span></big>'''<br><small>{{font|color=gray|nonlinearity 2}}</small>
|style="width:20%"|'''<big><span style="background:#ffb4b4">&nbsp; [[#wec E3 with 16 functions|E3]] &nbsp;</span></big>'''<br><small>{{font|color=gray|nonlinearity 2}}</small>
|-
|style="vertical-align: top;"| [[#wec_O_ordered_by_ggbec|by ''ggbec'']]<br><br>[[File:Venn 0000 0001.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0000 0111.svg|35px]]<br>''ggbec'' 1 with 64 functions<br><br>[[File:Venn 0001 0110.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0001 1001.svg|35px]]<br>''ggbec'' 22 with 64 functions
|style="vertical-align: top;"|  [[#wec_E0_ordered_by_ggbec|by ''ggbec'']]<br><br>[[File:Venn 0000 0000.svg|35px]]<br>''ggbec'' 0 with 2 functions<br><br>[[File:Venn 0000 1111.svg|35px]]<br>''ggbec'' 15 with 6 functions<br><br>[[File:Venn 0110 0110.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0110 1001.svg|35px]]<br>''ggbec'' 60 with 8 functions
|style="vertical-align: top;"|  [[#wec_E1_ordered_by_ggbec|by ''ggbec'']]<br><br>[[File:Venn 0001 0001.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0001 1110.svg|35px]]<br>''ggbec'' 3 with 48 functions
|style="vertical-align: top;"|  [[#wec_E2_ordered_by_ggbec|by ''ggbec'']]<br><br>[[File:Venn 0000 1001.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0101 0011.svg|35px]]<br>''ggbec'' 6 with 48 functions
| [[File:Venn 1000 0001.svg|35px]] &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [[File:Venn 0001 0111.svg|35px]]<br>''ggbec'' 23 with 16 functions
|-
|[[#wec_O_ordered_by_ggsec|by ''ggsec'']]
|[[#wec_E0_ordered_by_ggsec|by ''ggsec'']]
|[[#wec_E1_ordered_by_ggsec|by ''ggsec'']]
|[[#wec_E2_ordered_by_ggsec|by ''ggsec'']]
|-
|
<table class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto; border: none; border: 2px solid black;">
<tr><td rowspan="3" colspan="3"></td><th colspan="4">1</th><th colspan="4">22</th></tr>
<tr><th colspan="2">1</th><th colspan="2">7</th><th colspan="2">22</th><th colspan="2">25</th></tr>
<tr><th>1</th><th>127</th><th>7</th><th>31</th><th>22</th><th>107</th><th>25</th><th>61</th></tr>
<tr><th rowspan="4">1</th><th rowspan="2">1</th><th>1</th><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr>
<tr><th>127</th><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td><td></td><td></td></tr>
<tr><th rowspan="2">7</th><th>7</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td><td></td></tr>
<tr><th>31</th><td></td><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td></tr>
<tr><th rowspan="4">19</th><th rowspan="2">19</th><th>19</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td><td></td></tr>
<tr><th>55</th><td></td><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td></tr>
<tr><th rowspan="2">28</th><th>28</th><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td><td></td></tr>
<tr><th>61</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th rowspan="4">21</th><th rowspan="2">21</th><th>21</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td><td></td></tr>
<tr><th>87</th><td></td><td></td><td></td><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td><td></td><td></td></tr>
<tr><th rowspan="2">26</th><th>26</th><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td><td></td></tr>
<tr><th>91</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th rowspan="4">22</th><th rowspan="2">22</th><th>22</th><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td><td></td><td></td><td></td></tr>
<tr><th>107</th><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td><td></td><td></td></tr>
<tr><th rowspan="2">25</th><th>25</th><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td><td></td></tr>
<tr><th>103</th><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
</table>
|
<table class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto; border: none; border: 2px solid black;">
<tr><td rowspan="3" colspan="3"></td><th colspan="2">0</th><th>15</th><th colspan="2">60</th></tr>
<tr><th colspan="2">0</th><th>15</th><th>60</th><th>105</th></tr>
<tr><th>0</th><th>255</th><th>15</th><th>60</th><th>105</th></tr>
<tr><th rowspan="3">0</th><th rowspan="2">0</th><th>0</th><td style="background-color: #500;"><span style="color: white;">1</span></td><td></td><td></td><td></td><td></td></tr>
<tr><th>255</th><td></td><td style="background-color: #500;"><span style="color: white;">1</span></td><td></td><td></td><td></td></tr>
<tr><th>15</th><th>15</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">2</span></td><td></td><td></td></tr>
<tr><th rowspan="2">51</th><th>51</th><th>51</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">2</span></td><td></td><td></td></tr>
<tr><th>60</th><th>60</th><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">2</span></td><td></td></tr>
<tr><th rowspan="2">85</th><th>85</th><th>85</th><td></td><td></td><td style="background-color: #500;"><span style="color: white;">2</span></td><td></td><td></td></tr>
<tr><th>90</th><th>90</th><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">2</span></td><td></td></tr>
<tr><th rowspan="2">102</th><th>102</th><th>102</th><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">2</span></td><td></td></tr>
<tr><th>105</th><th>105</th><td></td><td></td><td></td><td></td><td style="background-color: black;"><span style="color: white;">2</span></td></tr>
</table>
|
<table class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto; border: none; border: 2px solid black;">
<tr><td rowspan="3" colspan="3"></td><th colspan="3">3</th></tr>
<tr><th colspan="2">3</th><th>30</th></tr>
<tr><th>3</th><th>63</th><th>30</th></tr>
<tr><th rowspan="2">3</th><th rowspan="2">3</th><th>3</th><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>63</th><td></td><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th rowspan="2">5</th><th rowspan="2">5</th><th>5</th><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>95</th><td></td><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th rowspan="3">17</th><th rowspan="2">17</th><th>17</th><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>119</th><td></td><td style="background-color: #500;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th>30</th><th>30</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th>54</th><th>54</th><th>54</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th>86</th><th>86</th><th>86</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
</table>
|
<table class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto; border: none; border: 2px solid black;">
<tr><td rowspan="3" colspan="3"></td><th colspan="3">6</th></tr>
<tr><th colspan="2">6</th><th>27</th></tr>
<tr><th>6</th><th>111</th><th>27</th></tr>
<tr><th rowspan="2">6</th><th rowspan="2">6</th><th>6</th><td style="background-color: black;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>111</th><td></td><td style="background-color: black;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th rowspan="3">18</th><th rowspan="2">18</th><th>18</th><td style="background-color: black;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>123</th><td></td><td style="background-color: black;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th>29</th><th>29</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th rowspan="3">20</th><th rowspan="2">20</th><th>20</th><td style="background-color: black;"><span style="color: white;">4</span></td><td></td><td></td></tr>
<tr><th>125</th><td></td><td style="background-color: black;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th>27</th><th>27</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
<tr><th>53</th><th>53</th><th>53</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">8</span></td></tr>
</table>
|
<table class="wikitable" style="text-align: center; margin-left: auto; margin-right: auto; border: none; border: 2px solid black;">
<tr><td rowspan="3" colspan="3"></td><th colspan="3">23</th></tr>
<tr><th>23</th><th colspan="2">24</th></tr>
<tr><th>23</th><th>24</th><th>126</th></tr>
<tr><th rowspan="3">23</th><th>23</th><th>23</th><td style="background-color: #500;"><span style="color: white;">8</span></td><td></td><td></td></tr>
<tr><th rowspan="2">24</th><th>24</th><td></td><td style="background-color: black;"><span style="color: white;">4</span></td><td></td></tr>
<tr><th>126</th><td></td><td></td><td style="background-color: black;"><span style="color: white;">4</span></td></tr>
</table>
|}

The tables above show how the equivalence classes are nested in each other. Each table shows the equivalence classes in a ''wec''.<br>
The vertical axes show ''ggsec''s, ''gsec''s and ''sec''s, and the horizontal ones show ''ggbec''s, ''gbec''s and ''bec''s.<br>
The entries in the matrices show which ''sec''s belong to which ''bec''s.<br>
The numbers in these fields show how many functions are in each ''sec''. A dark red background tells that the ''sec'' contains a monotonic function.

===''wec'' O with 128 functions===


====''wec'' O ordered by ''ggbec''====

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' O ordered by ''ggbec''
|-
|



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggbec'' 1 with 64 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 1 with 16 functions <span style="color: green; padding: 0 20px;">N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 1 with 8 functions
|-
| ''sec'' 1 with 8 functions<br><math>\scriptstyle A \land B \land C </math><br>[[File:Boolean functions like 1000 0000.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 127 with 8 functions
|-
| ''sec'' 127 with 8 functions<br><math>\scriptstyle A \lor B \lor C </math><br>[[File:Boolean functions like 1111 1110.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff"|Walsh spectra
|-
| [[File:0111 1111 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#d8d8d8" colspan="3"|''gbec'' 7 with 48 functions <span style="color: green; padding: 0 20px;">N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 7 with 24 functions
|-
|  ''sec'' 7 with 8 functions<br><math>\scriptstyle (A \lor B) \land C </math><br>[[File:Boolean functions like 1110 0000.svg|300px|]] 
| ''sec'' 19 with 8 functions<br><math>\scriptstyle (C \lor A) \land B </math><br>[[File:Boolean functions like 1100 1000.svg|300px|]] 
| ''sec'' 21 with 8 functions<br><math>\scriptstyle (B \lor C) \land A </math><br>[[File:Boolean functions like 1010 1000.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 31 with 24 functions
|-
| ''sec'' 31 with 8 functions<br><math>\scriptstyle (A \land B) \lor C </math><br>[[File:Boolean functions like 1111 1000.svg|300px|]] 
| ''sec'' 55 with 8 functions<br><math>\scriptstyle (C \land A) \lor B </math><br>[[File:Boolean functions like 1110 1100.svg|300px|]] 
| ''sec'' 87 with 8 functions<br><math>\scriptstyle (B \land C) \lor A </math><br>[[File:Boolean functions like 1110 1010.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:0111 0000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0100 1100 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0010 1010  Walsh spectrum.svg|thumb|center|230px|]]
|-
| [[File:1000 1111 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1011 0011 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1101 0101 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggbec'' 22 with 64 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 22 with 16 functions <span style="color: green; padding: 0 20px;">N(0)</span>
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 22 with 8 functions
|-
| ''sec'' 22 with 8 functions<br><math>\scriptstyle (A \lor B) \land (A \lor C) \land (B \lor C) \land (\neg A \lor \neg B \lor \neg C)</math><br>[[File:Boolean functions like 0110 1000.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 107 with 8 functions
|-
| ''sec'' 107 with 8 functions<br><math>\scriptstyle (A \land B) \lor (A \land C) \lor (B \land C) \lor (\neg A \land \neg B \land \neg C)</math><br>[[File:Boolean functions like 1110 1001.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff"|Walsh spectra
|-
| [[File:0001 0110 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#d8d8d8"|''gbec'' 25 with 48 functions <span style="color: green; padding: 0 20px;">N(0)</span>
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 25 with 24 functions
|-
| ''sec'' 25 with 8 functions<br><math>\scriptstyle (A \land B) \lor (\neg A \land \neg B \land C)</math><br>[[File:Boolean functions like 1001 1000.svg|300px|]] 
| ''sec'' 26 with 8 functions<br><math>\scriptstyle (C \land A) \lor (\neg C \land \neg A \land B)</math><br>[[File:Boolean functions like 1010 0100.svg|300px|]] 
| ''sec'' 28 with 8 functions<br><math>\scriptstyle (B \land C) \lor (\neg B \land \neg C \land A)</math><br>[[File:Boolean functions like 1100 0010.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 61 with 24 functions
|-
| ''sec'' 103 with 8 functions<br><math>\scriptstyle (A \lor B) \land (\neg A \lor \neg B \lor C)</math><br>[[File:Boolean functions like 1110 0110.svg|300px|]] 
|  ''sec'' 91 with 8 functions<br><math>\scriptstyle (C \lor A) \land (\neg C \lor \neg A \lor B)</math><br>[[File:Boolean functions like 1101 1010.svg|300px|]] 
|  ''sec'' 61 with 8 functions<br><math>\scriptstyle (B \lor C) \land (\neg B \lor \neg C \lor A)</math><br>[[File:Boolean functions like 1011 1100.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:0001 1001 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0010 0101 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0100 0011 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}

|}



|}

====''wec'' O ordered by ''ggsec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' O ordered by ''ggsec''
|-
|


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 1 with 32 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 1 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 1 with 8 functions<br><math>\scriptstyle A \land B \land C </math><br>[[File:Boolean functions like 1000 0000.svg|350px|]] 
| ''sec'' 127 with 8 functions<br><math>\scriptstyle A \lor B \lor C </math><br>[[File:Boolean functions like 1111 1110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 7 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 7 with 8 functions<br><math>\scriptstyle (A \lor B) \land C </math><br>[[File:Boolean functions like 1110 0000.svg|350px|]]  
| ''sec'' 31 with 8 functions<br><math>\scriptstyle (A \land B) \lor C </math><br>[[File:Boolean functions like 1111 1000.svg|350px|]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 1.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 19 with 32 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 28 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 28 with 8 functions<br><math>\scriptstyle (B \land C) \lor (A \land \neg B \land \neg C)</math><br>[[File:Boolean functions like 1100 0010.svg|350px|]] 
| ''sec'' 61 with 8 functions<br><math>\scriptstyle (B \lor C) \land (A \land \neg B \lor \neg C)</math><br>[[File:Boolean functions like 1011 1100.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 19 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 19 with 8 functions<br><math>\scriptstyle (A \lor C) \land B </math><br>[[File:Boolean functions like 1100 1000.svg|350px|]]  
| ''sec'' 55 with 8 functions<br><math>\scriptstyle (A \land C) \lor B </math><br>[[File:Boolean functions like 1110 1100.svg|350px|]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 19.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 21 with 32 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 26 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 26 with 8 functions<br><math>\scriptstyle (A \land C) \lor (\neg A \land B \land \neg C)</math><br>[[File:Boolean functions like 1010 0100.svg|350px|]] 
| ''sec'' 91 with 8 functions<br><math>\scriptstyle (A \lor C) \land (\neg A \lor B \lor \neg C)</math><br>[[File:Boolean functions like 1101 1010.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 21 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 21 with 8 functions<br><math>\scriptstyle (B \lor C) \land A </math><br>[[File:Boolean functions like 1010 1000.svg|350px|]]  
| ''sec'' 87 with 8 functions<br><math>\scriptstyle (B \land C) \lor A </math><br>[[File:Boolean functions like 1110 1010.svg|350px|]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 21.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 22 with 32 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 25 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 25 with 8 functions<br><math>\scriptstyle (A \land B) \lor (\neg A \land \neg B \land C)</math><br>[[File:Boolean functions like 1001 1000.svg|350px|]] 
| ''sec'' 103 with 8 functions<br><math>\scriptstyle (A \lor B) \land (\neg A \lor \neg B \lor C)</math><br>[[File:Boolean functions like 1110 0110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 22 with 16 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 22 with 8 functions<br><math>\scriptstyle (A \lor B) \land (A \lor C) \land (B \lor C) \land (\neg A \lor \neg B \lor \neg C)</math><br>[[File:Boolean functions like 0110 1000.svg|350px|]]  
| ''sec'' 107 with 8 functions<br><math>\scriptstyle (A \land B) \lor (A \land C) \lor (B \land C) \lor (\neg A \land \neg B \land \neg C)</math><br>[[File:Boolean functions like 1110 1001.svg|350px|]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 22.svg|600px]]
|}

|}



|}

===''wec'' E0 with 16 functions===


====''wec'' E0 ordered by ''ggbec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E0 ordered by ''ggbec''
|-
|



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggbec'' 0 = ''gbec'' 0 with 2 functions <span style="color: green; padding: 0 20px;">N(3, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 0 with 1 function
|-
| ''sec'' 0 with 1 function<br><math>\scriptstyle X \land \neg X</math><br>[[File:Boolean function 0000 0000.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4"|''bec'' 255 with 1 function
|-
| ''sec'' 255 with 1 function<br><math>\scriptstyle X \lor \neg X</math><br>[[File:Boolean function 1111 1111.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" |Walsh spectra
|-
| [[File:1111 1111 Walsh spectrum.svg|thumb|center|230px|]]
|}

|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;" colspan="3"|''ggbec'' 15 = ''gbec'' 15 = ''bec'' 15 with 6 functions <span style="color: green; padding: 0 20px;">N(2, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 15 with 2 functions<br><math>\scriptstyle C</math><br>[[File:Boolean functions like 1111 0000.svg|300px|]] 
| ''sec'' 51 with 2 functions<br><math>\scriptstyle B</math><br>[[File:Boolean functions like 1100 1100.svg|300px|]] 
| ''sec'' 85 with 2 functions<br><math>\scriptstyle A</math><br>[[File:Boolean functions like 1010 1010.svg|300px|]] 
|-
|colspan="3"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1111 0000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1100 1100 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1010 1010 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggbec'' 60 with 8 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="3"|''gbec'' 60 = ''bec'' 60 with 6 functions <span style="color: green; padding: 0 20px;">N(2, 2)</span>
|-
| ''sec'' 102 with 2 functions<br><math>\scriptstyle A \leftrightarrow B</math><br>[[File:Boolean functions like 1001 1001.svg|300px|]] 
| ''sec'' 90 with 2 functions<br><math>\scriptstyle A \leftrightarrow C</math><br>[[File:Boolean functions like 1010 0101.svg|300px|]] 
| ''sec'' 60 with 2 functions<br><math>\scriptstyle B \leftrightarrow C</math><br>[[File:Boolean functions like 1100 0011.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 105 = ''bec'' 105 with 2 functions <span style="color: green; padding: 0 20px;">N(2, 3)</span>
|-
| ''sec'' 105 with 2 functions<br><math>\scriptstyle A \oplus B \oplus C</math><br>[[File:Boolean functions like 1001 0110.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff" colspan="2"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|600px]]
| [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1001 1001 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1010 0101 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1100 0011 Walsh spectrum.svg|thumb|center|230px|]]
|-
| colspan="3"| [[File:1001 0110 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}



|}

====''wec'' E0 ordered by ''ggsec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E0 ordered by ''ggsec''
|-
|



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 0 with 4 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 0 with 2 functions <span style="color: green; padding: 0 20px;">15 in N(3, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 0 with 1 function<br><math>\scriptstyle X \land \neg X</math><br>[[File:Boolean function 0000 0000.svg|350px|]] 
| ''sec'' 255 with 1 function<br><math>\scriptstyle X \lor \neg X</math><br>[[File:Boolean function 1111 1111.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 15 with 2 functions <span style="color: green; padding: 0 20px;">4 in N(2, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 15 with 2 functions<br><math>\scriptstyle C</math><br>[[File:Boolean functions like 1111 0000.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 0.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 102 with 4 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 102 with 2 functions <span style="color: green; padding: 0 20px;">12 in N(2, 2)</span>
|-
| ''sec'' 102 with 2 functions<br><math>\scriptstyle A \leftrightarrow B</math><br>[[File:Boolean functions like 1001 1001.svg|350px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 105 with 2 functions <span style="color: green; padding: 0 20px;">10 in N(2, 3)</span>
|-
| ''sec'' 105 with 2 functions<br><math>\scriptstyle A \oplus B \oplus C</math><br>[[File:Boolean functions like 1001 0110.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 102.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 51 with 4 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 60 with 2 functions <span style="color: green; padding: 0 20px;">14 in N(2, 2)</span>
|-
| ''sec'' 60 with 2 functions<br><math>\scriptstyle B \leftrightarrow C</math><br>[[File:Boolean functions like 1100 0011.svg|350px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 51 with 2 functions <span style="color: green; padding: 0 20px;">7 in N(2, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 51 with 2 functions<br><math>\scriptstyle B</math><br>[[File:Boolean functions like 1100 1100.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 51.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 85 with 4 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 90 with 2 functions <span style="color: green; padding: 0 20px;">13 in N(2, 2)</span>
|-
| ''sec'' 90 with 2 functions<br><math>\scriptstyle A \leftrightarrow C</math><br>[[File:Boolean functions like 1010 0101.svg|350px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 85 with 2 functions <span style="color: green; padding: 0 20px;">9 in N(2, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 85 with 2 functions<br><math>\scriptstyle A</math><br>[[File:Boolean functions like 1010 1010.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 85.svg|600px]]
|}

|}



|}

===''wec'' E1 with 48 functions===


====''wec'' E1 ordered by ''ggbec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E1 = ''ggbec'' 3 with 48 functions
|-
|



{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 3 with 24 functions <span style="color: green; padding: 0 20px;">N(1, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 3 with 12 functions
|-
| ''sec'' 17 with 4 functions<br><math>\scriptstyle A \land B</math><br>[[File:Boolean functions like 1000 1000.svg|300px|]] 
| ''sec'' 5 with 4 functions<br><math>\scriptstyle A \land C</math><br>[[File:Boolean functions like 1010 0000.svg|300px|]] 
| ''sec'' 3 with 4 functions<br><math>\scriptstyle B \land C</math><br>[[File:Boolean functions like 1100 0000.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 63 with 12 functions
|-
| ''sec'' 119 with 4 functions<br><math>\scriptstyle A \lor B</math><br>[[File:Boolean functions like 1110 1110.svg|300px|]] 
| ''sec'' 95 with 4 functions<br><math>\scriptstyle A \lor C</math><br>[[File:Boolean functions like 1111 1010.svg|300px|]] 
| ''sec'' 63 with 4 functions<br><math>\scriptstyle B \lor C</math><br>[[File:Boolean functions like 1111 1100.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1000 1000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1010 0000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1100 0000 Walsh spectrum.svg|thumb|center|230px|]]
|-
| [[File:0111 0111 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0101 1111 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:0011 1111 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#d8d8d8" colspan="3"|''gbec'' 30 = ''bec'' 30 with 24 functions <span style="color: green; padding: 0 20px;">N(0)</span>
|-
| ''sec'' 30 with 8 functions<br><math>\scriptstyle (A \lor B) \leftrightarrow C</math><br>[[File:Boolean functions like 1110 0001.svg|300px|]] 
| ''sec'' 54 with 8 functions<br><math>\scriptstyle (A \lor C) \leftrightarrow B</math><br>[[File:Boolean functions like 1100 1001.svg|300px|]] 
| ''sec'' 86 with 8 functions<br><math>\scriptstyle (B \lor C) \leftrightarrow A</math><br>[[File:Boolean functions like 1010 1001.svg|300px|]] 
|-
| colspan="3"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1110 0001 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1100 1001 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1010 1001 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}



|}

====''wec'' E1 ordered by ''ggsec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E1 ordered by ''ggsec''
|-
|



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 17 with 16 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 17 with 8 functions <span style="color: green; padding: 0 20px;">5 in N(1, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 17 with 4 functions<br><math>\scriptstyle A \land B</math><br>[[File:Boolean functions like 1000 1000.svg|350px|]] 
| ''sec'' 119 with 4 functions<br><math>\scriptstyle A \lor B</math><br>[[File:Boolean functions like 1110 1110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 30 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 30 with 8 functions<br><math>\scriptstyle (A \lor B) \leftrightarrow C</math><br>[[File:Boolean functions like 1110 0001.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 17.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;" colspan="2"|''ggsec'' 3 = ''gsec'' 3 with 8 functions <span style="color: green; padding: 0 20px;">1 in N(1, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 3 with 4 functions<br><math>\scriptstyle B \land C</math><br>[[File:Boolean functions like 1100 0000.svg|350px|]] 
| ''sec'' 63 with 4 functions<br><math>\scriptstyle B \lor C</math><br>[[File:Boolean functions like 1111 1100.svg|350px|]]
|-
|colspan="2"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 3.svg|600px]]
|}
|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;" colspan="2"|''ggsec'' 5 = ''gsec'' 5 with 8 functions <span style="color: green; padding: 0 20px;">2 in N(1, 1)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 5 with 4 functions<br><math>\scriptstyle A \land C</math><br>[[File:Boolean functions like 1010 0000.svg|350px|]] 
| ''sec'' 95 with 4 functions<br><math>\scriptstyle A \lor C</math><br>[[File:Boolean functions like 1111 1010.svg|350px|]]
|-
|colspan="2"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 5.svg|600px]]
|}
|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 54 = ''gsec'' 54 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 54 with 8 functions<br><math>\scriptstyle (A \lor C) \leftrightarrow B</math><br>[[File:Boolean functions like 1100 1001.svg|350px|]] 
|-
|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 54.svg|600px]]
|}
|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 86 = ''gsec'' 86 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 86 with 8 functions<br><math>\scriptstyle (B \lor C) \leftrightarrow A</math><br>[[File:Boolean functions like 1010 1001.svg|350px|]] 
|-
|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 86.svg|600px]]
|}
|}



|}

===''wec'' E2 with 48 functions===


====''wec'' E2 ordered by ''ggbec''====

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E2 = ''ggbec'' 6 with 48 functions
|-
|



{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 6 with 24 functions <span style="color: green; padding: 0 20px;">N(1, 2)</span>
|-
|
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 6 with 12 functions
|-
| ''sec'' 6 with 4 functions<br><math>\scriptstyle (A \leftrightarrow B) \land C</math><br>[[File:Boolean functions like 1001 0000.svg|300px|]] 
| ''sec'' 18 with 4 functions<br><math>\scriptstyle (A \leftrightarrow C) \land B</math><br>[[File:Boolean functions like 1000 0100.svg|300px|]] 
| ''sec'' 20 with 4 functions<br><math>\scriptstyle (B \leftrightarrow C) \land A</math><br>[[File:Boolean functions like 1000 0010.svg|300px|]] 
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#f4f4f4" colspan="3"|''bec'' 111 with 12 functions
|-
| ''sec'' 111 with 4 functions<br><math>\scriptstyle (A \oplus B) \lor C</math><br>[[File:Boolean functions like 1111 0110.svg|300px|]] 
| ''sec'' 123 with 4 functions<br><math>\scriptstyle (A \oplus C) \lor B</math><br>[[File:Boolean functions like 1101 1110.svg|300px|]] 
| ''sec'' 125 with 4 functions<br><math>\scriptstyle (B \oplus C) \lor A</math><br>[[File:Boolean functions like 1011 1110.svg|300px|]] 
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1001 0000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1000 0100 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1000 0010 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#d8d8d8" colspan="3"|''gbec'' 27 = ''bec'' 27 with 24 functions <span style="color: green; padding: 0 20px;">N(0)</span>
|-
| ''sec'' 53 with 8 functions<br><math>\scriptstyle (A \land \neg C) \lor (B \land C)</math><br>[[File:Boolean functions like 1100 1010.svg|300px|]] 
| ''sec'' 29 with 8 functions<br><math>\scriptstyle (C \land \neg B) \lor (A \land B)</math><br>[[File:Boolean functions like 1011 1000.svg|300px|]] 
| ''sec'' 27 with 8 functions<br><math>\scriptstyle (B \land \neg A) \lor (C \land A)</math><br>[[File:Boolean functions like 1110 0100.svg|300px|]] 
|-
| colspan="3"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|600px]]
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa"
! bgcolor="#fff" colspan="3"|Walsh spectra
|-
| [[File:1100 1010 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1011 1000 Walsh spectrum.svg|thumb|center|230px|]]
| [[File:1110 0100 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}



|}

====''wec'' E2 ordered by ''ggsec''====


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E2 ordered by ''ggsec''
|-
|



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 20 with 16 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 20 with 8 functions <span style="color: green; padding: 0 20px;">8 in N(1, 2)</span>
|-
| ''sec'' 20 with 4 functions<br><math>\scriptstyle (B \leftrightarrow C) \land A</math><br>[[File:Boolean functions like 1000 0010.svg|350px|]] 
| ''sec'' 125 with 4 functions<br><math>\scriptstyle (B \oplus C) \lor A</math><br>[[File:Boolean functions like 1011 1110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 27 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 27 with 8 functions<br><math>\scriptstyle (B \land \neg A) \lor (C \land A)</math><br>[[File:Boolean functions like 1110 0100.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 20.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 18 with 16 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gsec'' 18 with 8 functions <span style="color: green; padding: 0 20px;">6 in N(1, 2)</span>
|-
| ''sec'' 18 with 4 functions<br><math>\scriptstyle (A \leftrightarrow C) \land B</math><br>[[File:Boolean functions like 1000 0100.svg|350px|]] 
| ''sec'' 123 with 4 functions<br><math>\scriptstyle (A \oplus C) \lor B</math><br>[[File:Boolean functions like 1101 1110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gsec'' 29 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 29 with 8 functions<br><math>\scriptstyle (C \land \neg B) \lor (A \land B)</math><br>[[File:Boolean functions like 1011 1000.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 18.svg|600px]]
|}

|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;" colspan="2"|''ggsec'' 6 = ''gsec'' 6 with 8 functions <span style="color: green; padding: 0 20px;">3 in N(1, 2)</span>
|-
| ''sec'' 6 with 4 functions<br><math>\scriptstyle (A \leftrightarrow B) \land C</math><br>[[File:Boolean functions like 1001 0000.svg|350px|]] 
| ''sec'' 111 with 4 functions<br><math>\scriptstyle (A \oplus B) \lor C</math><br>[[File:Boolean functions like 1111 0110.svg|350px|]]
|-
|colspan="2"|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 6.svg|600px]]
|}
|}



{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb" style="height: 28px;"|''ggsec'' 53 = ''gsec'' 53 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span>
|-
| ''sec'' 53 with 8 functions<br><math>\scriptstyle (A \land \neg C) \lor (B \land C)</math><br>[[File:Boolean functions like 1100 1010.svg|350px|]] 
|-
|
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Positions
|-
| [[File:3-ary Boolean functions; ggsec 53.svg|600px]]
|}
|}



|}

===''wec'' E3 with 16 functions===


{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#999" style="height: 35px;"|''wec'' E3 = ''ggbec'' 23 = ''ggsec'' 23 with 16 functions
|-
|

{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8" colspan="2"|''gbec'' 24 = ''gsec'' 24 with 8 functions  <span style="color: green; padding: 0 20px;">11 in N(1, 3)</span>
|-
| ''sec'' 24 with 4 functions<br><math>\scriptstyle (A \land B \land C) \lor (\neg A \land \neg B \land \neg C)</math><br>[[File:Boolean functions like 1000 0001.svg|350px|]] 
| ''sec'' 126 with 4 functions<br><math>\scriptstyle (A \lor B \lor C) \land (\neg A \lor \neg B \lor \neg C)</math><br>[[File:Boolean functions like 0111 1110.svg|350px|]]
|}
{| class="collapsible open" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#d8d8d8"|''gbec'' 23 = ''gsec'' 23 with 8 functions <span style="color: green; padding: 0 20px;">0 in N(0)</span> [[File:Tudor Rose.svg|20px]]
|-
| ''sec'' 23 with 8 functions<br><math>\scriptstyle (A \land B) \lor (A \land C) \lor (B \land C)</math><br><math>\scriptstyle (A \lor B) \land (A \lor C) \land (B \lor C)</math><br>[[File:Boolean functions like 1110 1000.svg|350px|]]  
|}
{| class="collapsible collapsed" style="text-align: center; width: 100%; border: 1px solid #aaa; border-bottom: 0"
! bgcolor="#fff" colspan="2"|Positions
|-
| [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|600px]]
| [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|600px]]
|}
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#fff"|Walsh spectra
|-
|
{|style="text-align: center; width:100%"
|[[File:1000 0001 Walsh spectrum.svg|thumb|center|230px|]]
|[[File:0111 1110 Walsh spectrum.svg|thumb|center|230px]]
|-
| colspan="2"|[[File:1110 1000 Walsh spectrum.svg|thumb|center|230px|]]
|}
|}

|}

==Monotonic functions==


<imagemap>
Image:Monotone Boolean functions 0,1,2,3.svg|400px|thumb|right|Formulas appear when the mouse is moved over the diagram, and the ''sec'' files are linked from them. In the ''sec'' files the monotonic functions are always in the top right position.

circle 659 623 30 [[File:Boolean function 0000 0000.svg|contradiction]]
circle 658 552 35 [[File:Boolean functions like 1000 0000.svg|A and B and C]]
circle 587 480 35 [[File:Boolean functions like 1000 1000.svg|A and B]]
circle 659 481 35 [[File:Boolean functions like 1010 0000.svg|A and C]]
circle 729 481 35 [[File:Boolean functions like 1100 0000.svg|B and C]]
circle 588 410 35 [[File:Boolean functions like 1010 1000.svg|(A and B) or (A and C)]]
circle 658 410 35 [[File:Boolean functions like 1100 1000.svg|(A and B) or (B and C)]]
circle 729 410 35 [[File:Boolean functions like 1110 0000.svg|(A and C) or (B and C)]]
circle 548 339 30 [[File:Boolean functions like 1010 1010.svg|A]]
circle 604 339 30 [[File:Boolean functions like 1100 1100.svg|B]]
circle 758 339 30 [[File:Boolean functions like 1111 0000.svg|C]]
circle 661 339 35 [[File:Boolean functions like 1110 1000.svg|(A or B) and (A or C) and (B or C) <====> (A and B) or (A and C) or (B and C)]]
circle 588 268 35 [[File:Boolean functions like 1110 1010.svg|(A or B) and (A or C)]]
circle 659 267 35 [[File:Boolean functions like 1110 1100.svg|(A or B) and (B or C)]]
circle 729 268 35 [[File:Boolean functions like 1111 1000.svg|(A or C) and (B or C)]]
circle 588 197 35 [[File:Boolean functions like 1110 1110.svg|A or B]]
circle 658 197 35 [[File:Boolean functions like 1111 1010.svg|A or C]]
circle 729 197 35 [[File:Boolean functions like 1111 1100.svg|B or C]]
circle 658 126 35 [[File:Boolean functions like 1111 1110.svg|A or B or C]]
circle 659 56 30 [[File:Boolean function 1111 1111.svg|tautology]]

desc bottom-left
</imagemap>


The right Hasse diagram in the file on the right shows the [[w:en:Dedekind number|20]] [[w:en:Monotonic function#Boolean functions|monotonic]] 3-ary Boolean functions.

If a ''sec'' has a monotonic function, all the ''sec''s in the ''gbec'' and ''gsec'' it belongs to also have one. So one can call ''sec''s, ''gbec''s and ''gsec''s monotonic, when they contain monotonic functions. There are 6 ''gbec''s and 12 ''gsec''s that are monotonic in this sense. Their collapsible boxes are marked with Tudor roses [[File:Tudor Rose.svg|20px]].

In the above tables that show the nesting of the equivalence classes the monotonic ''sec''s have a dark red instead of a black background.

{| class="wikitable"
! ''gbec'' !! weight !! ''gsec''
|-
| 0 
! 0, 8 
| 0
|-
| 1 
! 1, 7 
| 1
|-
| 3 
! 2, 6 
| 3, 5, 17
|-
| 7 
! 3, 5 
| 7, 19, 21
|-
| 15, 23 
! 4 
| 15, 23, 51, 85
|}

All ''wec''s except E2 contain monotonic functions.

<br style="clear: both;">

==Subgroups of nimber addition==


::''See: [[Subgroups of nimber addition#Z23|Subgroups of nimber addition#Z<sub>2</sub><sup>3</sup>]]''
[[oeis:A006116|16]] ''sec''s with in all 8 + 7*4 + 7*2 + 1 = 51 functions are related to subgroups of [[w:Nimber|nimber]] addition:

{| align="center"
|style="vertical-align:top;"|[[File:Z2^3; Lattice of subgroups Hasse diagram.svg|350px|thumb|These matrices [[w:Logical matrix|describe]] [[w:Equivalence relation|equivalence relations]].]]
|&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
|style="vertical-align:top;"|<imagemap>
File:Z2^3; Lattice of subgroups Hasse diagram (cubes).svg|350px|thumb|These cubes correspond to the matrices' top lines. In the linked files they are always in the bottom left position.
circle 341 1024 50 [[File:Boolean functions like 1000 0000.svg|1000 0000]]
circle 70 710 50 [[File:Boolean functions like 1100 0000.svg|1100 0000]]
circle 165 710 50 [[File:Boolean functions like 1010 0000.svg|1010 0000]]
circle 259 709 50 [[File:Boolean functions like 1000 1000.svg|1000 1000]]
circle 422 709 50 [[File:Boolean functions like 1000 0010.svg|1000 0010]]
circle 517 709 50 [[File:Boolean functions like 1000 0100.svg|1000 0100]]
circle 611 709 50 [[File:Boolean functions like 1001 0000.svg|1001 0000]]
circle 786 710 50 [[File:Boolean functions like 1000 0001.svg|1000 0001]]
circle 69 394 50 [[File:Boolean functions like 1010 1010.svg|1010 1010]]
circle 165 395 50 [[File:Boolean functions like 1100 1100.svg|1100 1100]]
circle 261 396 50 [[File:Boolean functions like 1111 0000.svg|1111 0000]]
circle 422 395 50 [[File:Boolean functions like 1100 0011.svg|1100 0011]]
circle 517 395 50 [[File:Boolean functions like 1010 0101.svg|1010 0101]]
circle 612 395 50 [[File:Boolean functions like 1001 1001.svg|1001 1001]]
circle 785 395 50 [[File:Boolean functions like 1001 0110.svg|1001 0110]]
circle 340 88 50 [[File:Boolean function 1111 1111.svg|1111 1111]]
desc none
</imagemap>
|}


In each [[Equivalence_classes_of_Boolean_functions#sec_matrix|''sec'' matrix]] its leading function appears in a ''sona'' pattern. This pattern is 0 in N(0) <abbr title="if and only if">iff</abbr> the ''sec'' contains 8 different functions. All the other different ''sona'' patterns appear only in the ''sec'' matrices of the actual ''sona''s between 1 and 15 and their Boolean complements. Each ''gbec'' is marked in green with its ''sona-bec'', and each ''gsec'' is marked with its ''sona-sec''.

==Complements in ''sec''s==


There are two kinds of ''sec''s that are their own complements (i.e. complete ''gsec''s): Those with ''sona'' rank 2, marked as N(2, ...), and the following seven ''sec''s containing 8 functions each:

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaa;"
! bgcolor="#bbb"| Complements in 8-element ''sec''s in a cube
|-
|

The position of the complement in the ''sec'' cube corresponds to the position of the ''sec'' cube in the big cube.

{{3-ary Boolean functions; complements in 8-element secs in a cube}}

|}

==''Gbec''s by similarity to 4-ary ''sec'' matrix==


In the octeract matrices of the ''gbec''s it can be seen, that they belong together in a way that is independent of ''ggbec''s and ''wec''s. Their pattern is part of a ''sec'' matrix of one of the five [[4-ary Boolean functions|4-ary]] [[w:Symmetric Boolean function|exact-value functions]]. On the left the functions in these ''gbecs'' are shown in the Hasse diagram.


===0===

{| class="wikitable" style="text-align: center;"
! [[File:3-ary Boolean functions in octeract graph; quaestor 00.svg|220px]]
! [[File:3-ary Boolean functions; quaestor 00.svg|200px]]<br>0
| [[File:3-ary Boolean functions; super-clan of Zhe 0.svg|200px]]<br>''gbec'' 0 [[File:Venn 0000 0000.svg|20px]] <small>''wec'' E0</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 6.svg|200px]]<br>''gbec'' 60 [[File:Venn 0110 0110.svg|20px]] <small>''wec'' E0</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 106.svg|200px]]<br>''gbec'' 24 [[File:Venn 1000 0001.svg|20px]] <small>''wec'' E3</small>
|}

===1===

{| class="wikitable" style="text-align: center;"
! [[File:3-ary Boolean functions in octeract graph; quaestor weight 1.svg|220px]]
! [[File:3-ary Boolean functions; quaestor weight 1.svg|200px]]<br>1, 2, 4, 8
| [[File:3-ary Boolean functions; super-clan of Zhe 128.svg|200px]]<br>''gbec'' 1 [[File:Venn 0000 0001.svg|20px]] <small>''wec'' O</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 134.svg|200px]]<br>''gbec'' 25 [[File:Venn 0001 1001.svg|20px]] <small>''wec'' O</small>
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#bbb"| ''gbec'' 1 and '''gbec''' 25 in a cube
|-
|
{{3-ary Boolean functions; gbec 1 and gbec 25 in a cube}}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa;"
! bgcolor="#bbb"| smallest 4-ary ''sec'' matrices
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|[[File:3-ary Boolean functions in octeract graph; quaestor 01.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 01.svg|87px]]<br>'''1'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 02.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 02.svg|87px]]<br>'''2'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 04.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 04.svg|87px]]<br>'''4'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 08.svg|250px]]<br>[[File:3-ary Boolean functions; related to 4-ary functions with weight 1; 08.svg|87px]]<br>'''8'''
|}
|}


===2===

{| class="wikitable" style="text-align: center;"
! [[File:3-ary Boolean functions in octeract graph; quaestor weight 2.svg|220px]]
! [[File:3-ary Boolean functions; quaestor weight 2.svg|200px]]<br>3, 5, 6, 9, 10, 12
| [[File:3-ary Boolean functions; super-clan of Zhe 8.svg|200px]]<br>''gbec'' 3 [[File:Venn 0001 0001.svg|20px]] <small>''wec'' E1</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 24.svg|200px]]<br>''gbec'' 30 [[File:Venn 0001 1110.svg|20px]] <small>''wec'' E1</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 40.svg|200px]]<br>''gbec'' 6 [[File:Venn 0000 1001.svg|20px]] <small>''wec'' E2</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 44.svg|200px]]<br>''gbec'' 27 [[File:Venn 0101 0011.svg|20px]] <small>''wec'' E2</small>
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#777"| ''gsec''s separately
|-
|
{| class="wikitable" style="text-align: center;"
!rowspan="2"| [[File:3-ary Boolean functions in octeract graph; quaestors 03 and 12.svg|220px]]
!rowspan="2" style="vertical-align:top;" | [[File:3-ary Boolean functions; quaestors 03 and 12.svg|200px]]<br>3, 12
| [[File:3-ary Boolean functions; super-family of Zhe 64.svg|200px]]<br>''gsec'' 3 [[File:Venn 0000 0011.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 66.svg|200px]]<br>''gsec'' 86 [[File:Venn 0101 0110.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 40.svg|200px]]<br>''gsec'' 20 [[File:Venn 0100 0001.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 44.svg|200px]]<br>''gsec'' 27 [[File:Venn 0010 0111.svg|20px]]
|-
|colspan="2"|<abbr title="binary Walsh spectrum">b.W.s.</abbr> 1 [[File:3-ary Boolean functions; binary Walsh spectrum 1.svg|29px]]
|colspan="2"|b.W.s. 6 [[File:3-ary Boolean functions; binary Walsh spectrum 6.svg|29px]]
|-
!rowspan="2"| [[File:3-ary Boolean functions in octeract graph; quaestors 05 and 10.svg|220px]]
!rowspan="2" style="vertical-align:top;" | [[File:3-ary Boolean functions; quaestors 05 and 10.svg|200px]]<br>5, 10
| [[File:3-ary Boolean functions; super-family of Zhe 32.svg|200px]]<br>''gsec'' 5 [[File:Venn 0000 0101.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 36.svg|200px]]<br>''gsec'' 54 [[File:Venn 0011 0110.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 72.svg|200px]]<br>''gsec'' 18 [[File:Venn 0010 0001.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 74.svg|200px]]<br>''gsec'' 29 [[File:Venn 0001 1101.svg|20px]]
|-
|colspan="2"|b.W.s. 2 [[File:3-ary Boolean functions; binary Walsh spectrum 2.svg|29px]]
|colspan="2"|b.W.s. 5 [[File:3-ary Boolean functions; binary Walsh spectrum 5.svg|29px]]
|-
!rowspan="2"| [[File:3-ary Boolean functions in octeract graph; quaestors 06 and 09.svg|220px]]
!rowspan="2" style="vertical-align:top;" | [[File:3-ary Boolean functions; quaestors 06 and 09.svg|200px]]<br>6, 9
| [[File:3-ary Boolean functions; super-family of Zhe 8.svg|200px]]<br>''gsec'' 17 [[File:Venn 0001 0001.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 24.svg|200px]]<br>''gsec'' 30 [[File:Venn 0001 1110.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 96.svg|200px]]<br>''gsec'' 6 [[File:Venn 0000 1001.svg|20px]]
| [[File:3-ary Boolean functions; super-family of Zhe 98.svg|200px]]<br>''gsec'' 53 [[File:Venn 0101 0011.svg|20px]]
|-
|colspan="2"|b.W.s. 4 [[File:3-ary Boolean functions; binary Walsh spectrum 4.svg|29px]]
|colspan="2"|b.W.s. 3 [[File:3-ary Boolean functions; binary Walsh spectrum 3.svg|29px]]
|}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa;"
! bgcolor="#bbb"| smallest 4-ary ''sec'' matrices
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|[[File:3-ary Boolean functions in octeract graph; quaestor 03.svg|250px]]<br>[[File:3-ary Boolean functions; related to 4-ary functions with weight 1; 03.svg|87px]]<br>'''3'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 05.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 05.svg|87px]]<br>'''5'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 06.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 06.svg|87px]]<br>'''6'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 09.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 09.svg|87px]]<br>'''9'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 10.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 10.svg|87px]]<br>'''10'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 12.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 12.svg|87px]]<br>'''12'''
|}
|}

===3===

{| class="wikitable" style="text-align: center;"
! [[File:3-ary Boolean functions in octeract graph; quaestor weight 3.svg|220px]]
! [[File:3-ary Boolean functions; quaestor weight 3.svg|200px]]<br>7, 11, 13, 14
| [[File:3-ary Boolean functions; super-clan of Zhe 130.svg|200px]]<br>''gbec'' 7 [[File:Venn 0000 0111.svg|20px]] <small>''wec'' O</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 150.svg|200px]]<br>''gbec'' 22 [[File:Venn 0001 0110.svg|20px]] <small>''wec'' O</small>
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa; border-bottom: 0;"
! bgcolor="#bbb"| ''gbec'' 7 and ''gbec'' 22 in a cube
|-
|
{{3-ary Boolean functions; gbec 7 and gbec 22 in a cube}}
|}

{| class="collapsible collapsed" style="width: 100%; text-align: center; border: 1px solid #aaa;"
! bgcolor="#bbb"| smallest 4-ary ''sec'' matrices
|-
|
{| class="wikitable" style="border-spacing: 0; margin: 1em auto; text-align: center;"
|[[File:3-ary Boolean functions in octeract graph; quaestor 07.svg|250px]]<br>[[File:3-ary Boolean functions; quaestor 07.svg|87px]]<br>'''7'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 11.svg|250px]]<br>[[File:3-ary Boolean functions; related to 4-ary functions with weight 1; 11.svg|87px]]<br>'''11'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 13.svg|250px]]<br>[[File:3-ary Boolean functions; related to 4-ary functions with weight 1; 13.svg|87px]]<br>'''13'''
|[[File:3-ary Boolean functions in octeract graph; quaestor 14.svg|250px]]<br>[[File:3-ary Boolean functions; related to 4-ary functions with weight 1; 14.svg|87px]]<br>'''14'''
|}
|}


===4===

{| class="wikitable" style="text-align: center;"
! [[File:3-ary Boolean functions in octeract graph; quaestor 15.svg|220px]]
! [[File:3-ary Boolean functions; quaestor 15.svg|200px]]<br>15
| [[File:3-ary Boolean functions; super-clan of Zhe 2.svg|200px]]<br>''gbec'' 15 [[File:Venn 0000 1111.svg|20px]] <small>''wec'' E0</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 22.svg|200px]]<br>''gbec'' 105 [[File:Venn 0110 1001.svg|20px]] <small>''wec'' E0</small>
| [[File:3-ary Boolean functions; super-clan of Zhe 104.svg|200px]]<br>''gbec'' 23 [[File:Venn 0001 0111.svg|20px]] <small>''wec'' E3</small>
|}

==References==


<references/>

[[Category:Studies of Boolean functions]]