View source for Linear and noble Boolean functions

{{Boolf header}}
<templatestyles src="Collapsible with classes/style.css" />

{| class="wikitable" style="float: right; text-align: center; "
! arity
! <math>n</math>
! 1
! 2
! 3
! 4 
! 5
|-
! linear
| <math>2^{n+1}</math>
| 4
| 8
| 16
| 32
| 64
|-
! noble
| <math>2^{2^{n-1}}</math>
| 2
| 4
| 16
| 256 
| 65536
|}

Among the truth tables for a given arity, the [[Linear Boolean functions|linears]] and the [[Noble Boolean functions|nobles]] are important subsets.

Each linear can be assigned a [[Noble Boolean functions#patrons|patron]], which is noble. Each noble can be assigned a prefect, which is linear.

For arity 3 they form a bijection. For higher arities the nobles outnumber the linears <small>(i.e. the patrons of the linears are a subset of the nobles)</small>.


==overview==

{| class="collapsible-with-classes collapsible collapsed wide followed"
!colspan="2"| 2-ary
|-
|
{{multiple image
 | align = center | total_width = 500
 | image1 = 2T linears with quadrants.svg
 | image2 = 2Z linears with quadrants.svg
 | footer = linears &nbsp; <small>(truth tables and Zhegalkin indices)</small>
}}
| [[File:2-ary nobles in matrix.svg|thumb|center|245px|nobles]]
|}

{| class="collapsible-with-classes collapsible open wide followed"
!colspan="2"| 3-ary
|-
|
{{multiple image
 | align = center | total_width = 700
 | image1 = 3T linears with quadrants.svg
 | image2 = 3Z linears with quadrants.svg
 | footer = linears &nbsp; <small>(truth tables and Zhegalkin indices)</small>
}}
| [[File:3-ary nobles in matrix.svg|thumb|center|345px|nobles]]
|}

{{Collapsible START|4-ary|collapsed wide}}
{{Collapsible START|linears &nbsp; <small>(truth tables)</small>|collapsed wide light followed}}
[[File:4T linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|linears &nbsp; <small>(Zhegalkin indices)</small>|collapsed wide light followed}}
[[File:4Z linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|nobles|collapsed wide light}}
[[File:4-ary nobles in matrix.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible END}}

==3-ary==

{| class="collapsible-with-classes collapsible collapsed wide light gap-below"
!colspan="3" style="background-color: #f0f0f0; color: gray;" | nobles in tesseract
|-
| [[File:3-ary nobles in tesseract.svg|500px]]
| [[File:2T principality; faction size 1; king index 0.svg|thumb|center|280px|dihedral symmetry]]
| [[File:2T principality; faction size 3; king index 2.svg|thumb|center|280px|mirror symmetry]]
|-
|}

{| class="collapsible-with-classes collapsible open wide followed"
!colspan="2" bgcolor="#ddd" | quadrants
|-
|style="padding-right: 15px;"| [[File:3-ary linear to noble, quadrant 0.svg|thumb|right|300px| {{colorbox|#e30000}} &nbsp; 0 &nbsp; <small>(even, evil)</small>]]
|style="padding-left: 15px;"| [[File:3-ary linear to noble, quadrant 3.svg|thumb|left|300px| {{colorbox|#00cc00}} &nbsp; 3 &nbsp; <small>(odd, odious)</small>]]
|-
|style="padding-right: 15px;"| [[File:3-ary linear to noble, quadrant 2.svg|thumb|right|300px| {{colorbox|#ffb000}} &nbsp; 2 &nbsp; <small>(even, odious)</small>]]
|style="padding-left: 15px;"| [[File:3-ary linear to noble, quadrant 1.svg|thumb|left|300px| {{colorbox|#0055ff}} &nbsp; 1 &nbsp; <small>(odd, evil)</small>]]
|}

{| class="collapsible-with-classes collapsible collapsed wide followed"
!colspan="2" bgcolor="#ddd" | halves
|-
|style="padding-right: 15px;"| [[File:3-ary linear to noble, even.svg|thumb|right|300px|even<br><small>Walsh functions</small>]]
|style="padding-left: 15px;"| [[File:3-ary linear to noble, odd.svg|thumb|left|300px|odd<br><small>complements of Walsh functions</small>]]
|}

{| class="collapsible-with-classes collapsible collapsed wide"
!colspan="2" bgcolor="#ddd" | all
|-
| [[File:3-ary noble to linear.svg|300px|center]]
|}


==4-ary==

===linear to patron (noble)===

There are 32 linears, which form 10 factions. <small>(Even and odd faction for each Walsh weight 0...4.)</small> Their patrons are 32 nobles, which also form 10 factions <small>(among the 44 noble factions)</small>.

Their [[Boolf-term#junior|juniors]] are the 3-ary Boolean functions with consul 0. &nbsp; <small style="opacity: .5;">(This is not the case for arities 2 and 3, but probably for all arities ≥ 4.)</small>

{{Collapsible START|list|collapsed wide followed}}
The faction is determined by Walsh weight and parity (or quadrant), and represented by variants of the quadrant colors.<br>
<small>A good sort order is first by quadrant, and [[w:Help:Sortable tables#Secondary key|then]] by Walsh weight.</small>
{{Linear to patron 4-ary list}}
{{Collapsible START|sequences|collapsed wide light}}
{{Linear to patron 4-ary sequences}}
{{Collapsible END}}
{{Collapsible END}}

{{Collapsible START|details for each faction|collapsed wide followed}}
The numbers next to the images correspond to the truth tables.<br>
The linears are in the left columns, their patrons in the middle, and their [[Zhegalkin twins|twins]] on the right. &nbsp;
<small>(So the [[Zhegalkin matrix|Zhegalkin indices]] of the linear functions are in the columns on the right.)</small>
{{Linear to patron 4-ary factions}}
{{Collapsible END}}

{{Collapsible START|3-ary noble indices &nbsp; <small>(juniors)</small>|collapsed wide}}
{| style="text-align: center; width: 100%;"
|colspan="2" style="text-align: left;"|
The factions are shown in ten different colors, which are variants of the quadrant colors. &nbsp;
<small>(There are three shades of red and green, and two shades of blue and yellow.)</small>
|-
| [[File:3T linear to patron.svg|thumb|center|500px|truth tables]]
| [[File:3Z linear to patron.svg|thumb|center|500px|Zhegalkin indices]]
|}
{{Collapsible END}}

The following images show that pairs of complementary factions are in the same principality.

{{Collapsible START|quadrants 0 and 3|collapsed wide followed}}
{| class="wikitable" style="float: right; text-align: center;"
|- style="font-size: 65%;"
! king
| 0
| 3808
| 26752
|-
! king index
| 0
| 14
| 104
|}
{{colorbox|#e30000}} &nbsp; The patrons of the 8 even Walsh functions <small>(quadrant 0)</small> are the red entries in these 3 principalities.<br>
{{colorbox|#00cc00}} &nbsp; The patrons of their complements <small>(quadrant 3)</small> are the green entries.
{| class="collapsible-with-classes collapsible collapsed light wide followed" style="text-align: center;"
!colspan="3"| truth tables
|-
| [[File:3T principality; faction size 1; king index 0.svg|400px]]
| [[File:3T principality; faction size 6; king index 14.svg|400px]]
| [[File:3T principality; faction size 1; king index 104.svg|400px]]
|}
{| class="collapsible-with-classes collapsible collapsed light wide" style="text-align: center;"
!colspan="3"| Zhegalkin indices
|-
| [[File:3Z principality; faction size 1; king index 0.svg|400px]]
| [[File:3Z principality; faction size 6; king index 14.svg|400px]]
| [[File:3Z principality; faction size 1; king index 104.svg|400px]]
|}
{{Collapsible END}}

{{Collapsible START|quadrants 1 and 2|collapsed wide}}
{| class="wikitable" style="float: right; text-align: center;"
|- style="font-size: 65%;"
! king
| 5760
| 10920
|-
! king index
| 22
| 42
|}
{{colorbox|#ffb000}} &nbsp; The patrons of the 8 odious Walsh functions <small>(quadrant 2)</small> are the yellow entries in these 2 principalities.<br>
{{colorbox|#0055ff}} &nbsp; The patrons of their complements <small>(quadrant 1)</small> are the blue entries.
{| class="collapsible-with-classes collapsible collapsed light wide followed" style="text-align: center;"
!colspan="3"| truth tables
|-
| [[File:3T principality; faction size 4; king index 22.svg|400px]]
| [[File:3T principality; faction size 4; king index 42.svg|400px]]
|}
{| class="collapsible-with-classes collapsible collapsed light wide" style="text-align: center;"
!colspan="3"| Zhegalkin indices
|-
| [[File:3Z principality; faction size 4; king index 22.svg|400px]]
| [[File:3Z principality; faction size 4; king index 42.svg|400px]]
|}
{{Collapsible END}}

===noble to prefect (linear)===

There are 256 nobles. A prefect is one of the 32 linears. Every linear is the prefect of 8 nobles.

{{Collapsible START|list &nbsp; <small>(32 × 8 matrix)</small>|collapsed wide followed}}
This table is an extension of the one shown above (from linear to patron). The leftmost noble column is equal to that of patrons.<br>
But here the assignment goes in the other direction. Each of the nobles on the right has the prefect on the left. <small>E.g. the nobles 3870 and 2600 have prefect Ж 17.</small>

The nobles are represented by the integer values of their truth tables, which are also their Zhegalkin indices.<br>
The small numbers to their right are the juniors 0...255.<br>
<small>They correspond to 3-ary Boolean functions, so among them are also linears and nobles. The linears are highlighted in bold, and the nobles with a box.</small>

The background colors of the nobles denote the [[Noble Boolean functions#quadrants|noble quadrants]].<br>
The tiny red numbers are the king indices. Together with the noble quadrants they denote the noble factions. &nbsp; <small opacity: .5;>(The eleven king indices are 0, 2, 6, 8, 14, 22, 26, 42, 44, 104, 110.)</small>

{{Noble to prefect 4-ary}}
{{Collapsible END}}

{{Collapsible START|3-ary subprefects &nbsp; <small>(16 × 16 matrices)</small>|collapsed wide}}
The right side of the following table shows the same information as the table above.<br>
Entries of one color in one matrix are the eight juniors in a row of the table above. &nbsp; <small>(Linears are highlighted in bold, and the nobles with a crown.)</small>

The left side of the table shows the prefects of all 3-ary Boolean functions. <small>(And the great prefects as sort keys.)</small><br>
It can be seen, that the {{w|Image (mathematics)|preimages}} of the 4-ary prefects are a refinement of the 3-ary prefects. This motivates the term ''subprefect''.

{{Noble to prefect 4-ary matrices}}

<small>A more detailed version of this table can be seen [[commons:Category:3-ary Boolean functions in octeract matrix; great subprefects|here]].</small>
{{Collapsible END}}


[[Category:Linear and noble Boolean functions]]