Template:Zhegalkin matrix/matrix pi xor phi

Template:Collapsible START Let $\pi ={\mathrm{\Pi }}_2$ (length 16), $\varphi ={\mathrm{\Phi }}_3$ (length 16) and $\xi ={\mathrm{\Xi }}_3$ (length 256).
Let $I_k=\{i\mid {\xi }_i={\varphi }_k\}$ be the set of places where $\xi$ has the entry ${\varphi }_k$.

$\pi$ is the left column of the following matrix.
$\varphi$ is its top row, and also shown in the column to its right.
The matrix entries are the bitwise XORs of its left column and top row.
$I_k$ is row $k$ as a set of numbers: $\{{\pi }_k\oplus f\mid f\in \varphi \}$

Example:   In which places is $\xi$ equal to ${\varphi }_{10}=168$?
${\pi }_{10}=2⟹I_{10}=\{2\oplus f\mid f\in \varphi \}=\{2,28,...,226,252\}$
It can be easily seen, that the first and the last entry 168 in $\xi$ is in places 2 and 252.

0	0	30	40	54	72	86	96	126	128	158	168	182	200	214	224	254		0
1	15	17	39	57	71	89	111	113	143	145	167	185	199	217	239	241		30
2	10	20	34	60	66	92	106	116	138	148	162	188	194	220	234	244		40
3	5	27	45	51	77	83	101	123	133	155	173	179	205	211	229	251		54
4	12	18	36	58	68	90	108	114	140	146	164	186	196	218	236	242		72
5	3	29	43	53	75	85	99	125	131	157	171	181	203	213	227	253		86
6	6	24	46	48	78	80	102	120	134	152	174	176	206	208	230	248		96
7	9	23	33	63	65	95	105	119	137	151	161	191	193	223	233	247		126
8	8	22	32	62	64	94	104	118	136	150	160	190	192	222	232	246		128
9	7	25	47	49	79	81	103	121	135	153	175	177	207	209	231	249		158
10	2	28	42	52	74	84	98	124	130	156	170	180	202	212	226	252		168
11	13	19	37	59	69	91	109	115	141	147	165	187	197	219	237	243		182
12	4	26	44	50	76	82	100	122	132	154	172	178	204	210	228	250		200
13	11	21	35	61	67	93	107	117	139	149	163	189	195	221	235	245		214
14	14	16	38	56	70	88	110	112	142	144	166	184	198	216	238	240		224
15	1	31	41	55	73	87	97	127	129	159	169	183	201	215	225	255		254

The lower 8×16 matrix corresponds to row 10.
The one above contains the Zhegalkin indices, and has the same set of columns.

Nobles for comparison