Noble Boolean functions

Studies of Boolean functions

Ж

ANF   matrix   twins

noble   gentle

smallest Zhegalkin index

Noble Boolean functions are those who are their own Zhegalkin twins, i.e. the binary expression of their ANF is equal to their truth table.
They correspond to fixed points in the Zhegalkin permutation.
They are all even, i.e. the first digit of their truth table is false.
When a Boolean function is noble, its whole faction is noble.
Within the this project it is slightly misleading to apply the term noble to Boolean functions. It is a property of a truth table with a specific length.

$\Pi$           A197819

Fixed points are in parentheses. k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n 0 (0) (1) 1 (0) 3 (2) 1 2 (0) 15 10 5 12 3 (6) 9 (8) 7 2 13 4 11 (14) 1 row 3 of length 256:     0, 255, 170, 85 ... 84, 171, 254, 1 [0, 255, 170, 85, 204, 51, 102, 153, 136, 119, 34, 221, 68, 187, 238, 17, 240, 15, 90, 165, 60, 195, 150, 105, 120, 135, 210, 45, 180, 75, 30, 225, 160, 95, 10, 245, 108, 147, 198, 57, 40, 215, 130, 125, 228, 27, 78, 177, 80, 175, 250, 5, 156, 99, 54, 201, 216, 39, 114, 141, 20, 235, 190, 65, 192, 63, 106, 149, 12, 243, 166, 89, 72, 183, 226, 29, 132, 123, 46, 209, 48, 207, 154, 101, 252, 3, 86, 169, 184, 71, 18, 237, 116, 139, 222, 33, 96, 159, 202, 53, 172, 83, 6, 249, 232, 23, 66, 189, 36, 219, 142, 113, 144, 111, 58, 197, 92, 163, 246, 9, 24, 231, 178, 77, 212, 43, 126, 129, 128, 127, 42, 213, 76, 179, 230, 25, 8, 247, 162, 93, 196, 59, 110, 145, 112, 143, 218, 37, 188, 67, 22, 233, 248, 7, 82, 173, 52, 203, 158, 97, 32, 223, 138, 117, 236, 19, 70, 185, 168, 87, 2, 253, 100, 155, 206, 49, 208, 47, 122, 133, 28, 227, 182, 73, 88, 167, 242, 13, 148, 107, 62, 193, 64, 191, 234, 21, 140, 115, 38, 217, 200, 55, 98, 157, 4, 251, 174, 81, 176, 79, 26, 229, 124, 131, 214, 41, 56, 199, 146, 109, 244, 11, 94, 161, 224, 31, 74, 181, 44, 211, 134, 121, 104, 151, 194, 61, 164, 91, 14, 241, 16, 239, 186, 69, 220, 35, 118, 137, 152, 103, 50, 205, 84, 171, 254, 1]

$\Phi _{n}~~=~~\{i\mid \Pi _{n,~i}=i\}$           A358167

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n 0 0 1 0 2 2 0 6 8 14 3 0 30 40 54 72 86 96 126 128 158 168 182 200 214 224 254 row 4 of length 256:     0, 510, 680, 854 ... 64680, 64854, 65024, 65534 [0, 510, 680, 854, 1224, 1334, 1632, 1950, 2176, 2430, 2600, 3030, 3144, 3510, 3808, 3870, 4320, 4382, 4680, 5046, 5160, 5590, 5760, 6014, 6240, 6558, 6856, 6966, 7336, 7510, 7680, 8190, 8320, 8574, 8744, 9174, 9288, 9654, 9952, 10014, 10240, 10750, 10920, 11094, 11464, 11574, 11872, 12190, 12384, 12702, 13000, 13110, 13480, 13654, 13824, 14334, 14560, 14622, 14920, 15286, 15400, 15830, 16000, 16254, 16512, 16766, 16936, 17366, 17480, 17846, 18144, 18206, 18432, 18942, 19112, 19286, 19656, 19766, 20064, 20382, 20576, 20894, 21192, 21302, 21672, 21846, 22016, 22526, 22752, 22814, 23112, 23478, 23592, 24022, 24192, 24446, 24576, 25086, 25256, 25430, 25800, 25910, 26208, 26526, 26752, 27006, 27176, 27606, 27720, 28086, 28384, 28446, 28896, 28958, 29256, 29622, 29736, 30166, 30336, 30590, 30816, 31134, 31432, 31542, 31912, 32086, 32256, 32766, 32768, 33278, 33448, 33622, 33992, 34102, 34400, 34718, 34944, 35198, 35368, 35798, 35912, 36278, 36576, 36638, 37088, 37150, 37448, 37814, 37928, 38358, 38528, 38782, 39008, 39326, 39624, 39734, 40104, 40278, 40448, 40958, 41088, 41342, 41512, 41942, 42056, 42422, 42720, 42782, 43008, 43518, 43688, 43862, 44232, 44342, 44640, 44958, 45152, 45470, 45768, 45878, 46248, 46422, 46592, 47102, 47328, 47390, 47688, 48054, 48168, 48598, 48768, 49022, 49280, 49534, 49704, 50134, 50248, 50614, 50912, 50974, 51200, 51710, 51880, 52054, 52424, 52534, 52832, 53150, 53344, 53662, 53960, 54070, 54440, 54614, 54784, 55294, 55520, 55582, 55880, 56246, 56360, 56790, 56960, 57214, 57344, 57854, 58024, 58198, 58568, 58678, 58976, 59294, 59520, 59774, 59944, 60374, 60488, 60854, 61152, 61214, 61664, 61726, 62024, 62390, 62504, 62934, 63104, 63358, 63584, 63902, 64200, 64310, 64680, 64854, 65024, 65534]

3-ary nobles (Venn diagrams) assigned to juniors (tesseract vertices)

3-ary and 4-ary nobles assigned to juniors (in square matrices)
These matrices show the 2-ary and 3-ary Boolean functions, represented by the big gray integers (corresponding to the truth tables). The small black integers above them are the senior nobles (i.e. one arity above). (The highlighted big numbers are the king indices.) 3-ary nobles 4-ary nobles

illustration of fixed points $\Phi _{3}$ in permutation $\Pi _{3}$

$\Phi _{3}$ as 8×16 matrix          $\Phi _{4}$ as 16×256 matrix

row 3 row 4 The images show how $\Phi _{n}$ is derived from $\Pi _{n-1}$, the permutation of the same length. The long matrices have two halves: In the upper half the bit pattern in column $k$ is that of $\Xi _{n-1,~k}~=~\Pi _{n-1,~k}\oplus k$.    (Compare triangle Ξ in the next section.) The bit pattern in the lower half is identical to that of the gray column indices. $\Phi _{n,~k}~~=~~\Xi _{n-1,~k}+{\bigl (}2^{2^{n-1}}\cdot k{\bigr )}~~=~~{\bigl (}\Pi _{n-1,~k}\oplus k{\bigr )}+{\bigl (}2^{2^{n-1}}\cdot k{\bigr )}$ The small matrices on the left are divided in the same way: The upper half is a Sierpiński triangle without the main diagonal, and the lower half is the main diagonal. These properties of noble Boolean functions can be derived from this: They are even, i.e. place 0 is always false. The 1-bit places (e.g. 1, 2, 4, 8) have the same truth value. (Those where it is false/true shall be called weak/strong.) Half of them are evil/odious, which is indicated by the last place being false/true. (The odious ones are on the right, just like in triangle Π.)

row sums of $\Phi _{n}$

The row sum of $\Phi _{n}$ for $n\geq 1$ is $~~~2^{a}\cdot (2^{b}-1)~~~=~~~2^{a+b}-2^{a}~~~$ with $~~~a=2^{n-1}~~~$ and $~~~b=2^{n}-1~~~$. That is a binary number consisting of $a$ 0s at the little and $b$ 1s at the big end. E.g. that of $\Phi _{2}$ is 0 + 6 + 8 + 14 = 28. (11100 as a binary number.) n sum of row n 0 1 1 $2^{1}\cdot (2^{1}-1)~~~=~~~2^{2}-2^{1}$ 2 2 $2^{2}\cdot (2^{3}-1)~~~=~~~2^{5}-2^{2}$ 28 3 $2^{4}\cdot (2^{7}-1)~~~=~~~2^{11}-2^{4}$ 2032 4 $2^{8}\cdot (2^{15}-1)~~~=~~~2^{23}-2^{8}$ 8388352 5 $2^{16}\cdot (2^{31}-1)~~~=~~~2^{47}-2^{16}$ 140737488289792

quadrants

It is easily seen, that the left and right half of each row differ only in the last digit.
Those on the left/right have even/odd weight. They shall be called evil/odious.

half rows

There are two ways to derive one half from the other. Let $L$ be the left (evil) and $R$ be the right (odious) half. Let $p=2^{(2^{n}-1)}$ be the unique power of two, and $q=2^{2^{n}}-2$ be the highest entry.   (They are the first and last entries of $R$.) Let $j$ be the mirrored index of $i$.   ($j$ is as far from the right as $i$ is from the left.) Then $R_{i}~=~L_{i}+p~=~q-L_{j}$.     (Bitwise XOR can be used instead of plus and minus.) The following Python code illustrates this for rows 1 to 3: left_and_right_half = { 1: [ [0], [2] ], 2: [ [0, 6], [8, 14] ], 3: [ [ 0, 30, 40, 54, 72, 86, 96, 126], [128, 158, 168, 182, 200, 214, 224, 254] ] } for n, (left, right) in left_and_right_half.items(): length = 2 ** (2 ** (n - 1) - 1) p = 2 ** (2 ** n - 1) q = 2 ** (2 ** n) - 2 for i in range(length): j = length - i - 1 assert right[i] == left[i] + p == q - left[j] assert right[i] == left[i] ^ p == q ^ left[j]

There is a second way to partition the nobles in two halves:
Those in even/odd places of the triangle row have false/true entries in all 1-bit places of their truth table. They shall be called weak/strong.

So the nobles can be partitioned into four quadrants by depravity and strength.

illustration of quadrants for n = 3
Vertically adjacent quadrants contain relative complements. Horizontally adjacent quadrants differ only in the central vertex. (40 and 214 are relative complements. 40 and 168 differ only in the central vertex.) The 16 noble 3-ary Boolean functions form 8 factions. So there are 2 royal factions.

Nobles that are evil and weak shall be called royal.
Each noble corresponds to a royal, and can easily be derived from it. (A royal corresponds to itself.)
When a Boolean function is royal, its whole faction is royal.
The function with the smallest Zhegalkin index in a faction shall be called king, and be used to represent it.
So all nobles of a given arity can effectively be represented by a rather short list of kings.

triangle of royals

0 1 2 3 [0] [0] [0] [0, 40, 72, 96] row 4 of length 64:     0, 680, 1224, 1632 ... 30816, 31432, 31912, 32256 [0, 680, 1224, 1632, 2176, 2600, 3144, 3808, 4320, 4680, 5160, 5760, 6240, 6856, 7336, 7680, 8320, 8744, 9288, 9952, 10240, 10920, 11464, 11872, 12384, 13000, 13480, 13824, 14560, 14920, 15400, 16000, 16512, 16936, 17480, 18144, 18432, 19112, 19656, 20064, 20576, 21192, 21672, 22016, 22752, 23112, 23592, 24192, 24576, 25256, 25800, 26208, 26752, 27176, 27720, 28384, 28896, 29256, 29736, 30336, 30816, 31432, 31912, 32256]

4-ary royals in 16×64 matrix
The columns of this matrix are the 4-ary royal Boolean functions. Compare $\Phi _{4}$ as 16×256 matrix, shown above. The 16×6 matrix on the left is shown as the 16×8 matrix from that file, with the left and right column blacked out.

triangle of kings

Number of royal factions for arities 0...5:     1, 1, 1, 2, 11, 272 This triangle shows the representatives of each royal faction. 0 1 2 3 4 [0] [0] [0] [0, 40] [0, 680, 1632, 2176, 3808, 5760, 6856, 10920, 11464, 26752, 28384] row 5 of length 272:     0, 174760, 419424 ... 2091314912, 2091559592, 2093664936 [0, 174760, 419424, 559232, 664104, 978656, 1480320, 1603680, 1757896, 2631680, 2785960, 2942152, 3034720, 4112000, 6875264, 7245536, 8289792, 8421376, 8840800, 9085480, 10025056, 10395136, 11053056, 11456096, 16711168, 18245760, 18384480, 18492616, 18800640, 19453440, 19561640, 19837024, 20769320, 23772200, 25067520, 25351336, 25753216, 26667136, 26913992, 27983016, 36208640, 36317864, 36457672, 36628064, 37534760, 37775488, 37929512, 38330368, 38700640, 39081128, 39255552, 39500360, 39640104, 39745152, 40561192, 40668232, 41712808, 41866752, 42107560, 42263752, 42508416, 42911456, 43433600, 43711176, 43818152, 44739240, 44879048, 46196864, 46844584, 47502504, 48166528, 50134184, 54454400, 56159968, 56526976, 57196160, 57596128, 58491560, 58775040, 59158624, 59438120, 60090920, 60337760, 60737536, 61406720, 62702120, 63232584, 63895648, 107129512, 107513984, 107933408, 108435072, 108558432, 108712648, 109109248, 109479520, 109756968, 110279240, 110663776, 111083008, 111340072, 111724544, 112111200, 112242784, 112645632, 113046568, 115795552, 116979808, 125339232, 125755392, 126006440, 126391936, 126808288, 126938848, 127585480, 127975040, 128375008, 128776904, 129164488, 132707968, 142781096, 143165568, 143584992, 143716576, 144086656, 144364232, 144744576, 145147616, 145548488, 145947336, 146331872, 146718336, 149481600, 150402688, 160605256, 161406976, 161674280, 162059776, 162443360, 163375656, 166131808, 178430504, 178570312, 179234400, 179370080, 180017736, 180141096, 180936704, 181861888, 182246440, 183167528, 183825448, 197060736, 198766304, 199555784, 250539744, 250671328, 375941248, 376095272, 377421312, 377662120, 378046592, 378466016, 378967680, 379245256, 380429512, 381599360, 384362624, 394329600, 394696800, 394976296, 396288000, 398256680, 411099136, 411469408, 411746856, 412269128, 412653664, 413072896, 413329960, 413714432, 414101088, 414232672, 414635520, 415036456, 417785440, 418969696, 428928736, 429575368, 429964928, 430364896, 430766792, 431154376, 434697856, 446752896, 447400616, 447524040, 447918792, 448058536, 448722560, 449244896, 450024136, 450425056, 451208392, 452655816, 464578144, 467066880, 468382760, 518053888, 520027648, 679510016, 679929440, 680174120, 680313928, 680836136, 681113696, 682141696, 682544736, 683862568, 684247040, 685168128, 685569064, 687799808, 697755776, 698002632, 698141352, 699071656, 700366976, 700897504, 701299400, 715827880, 715967688, 717285504, 717933224, 718591144, 719255168, 721222824, 733790760, 734321224, 734984288, 751234784, 751619272, 752014024, 752414944, 753198280, 754645704, 769056768, 770372648, 786884192, 788068448, 1019396648, 1019797504, 1020722688, 1022028328, 1037627104, 1055451264, 1753393832, 1753778304, 1754197728, 1754699392, 1754976968, 1756161224, 1757331072, 1760094336, 1772019712, 1773988392, 1789847136, 1791549440, 1793780264, 1807673472, 1809379040, 1861152480] Finding the next row would require looping through the $2^{30}$ (slightly more than a billion) 6-ary royal Boolean functions. triangle of king indices These are the index numbers of the kings in the respective sequence of nobles. It seems, that each row is a subset of the next. 0 1 2 3 4 [0] [0] [0] [0, 2] [0, 2, 6, 8, 14, 22, 26, 42, 44, 104, 110] row 5 of length 272:     0, 2, 6 ... 27582, 27608, 28398 [0, 2, 6, 8, 10, 14, 22, 24, 26, 40, 42, 44, 46, 62, 104, 110, 126, 128, 134, 138, 152, 158, 168, 174, 254, 278, 280, 282, 286, 296, 298, 302, 316, 362, 382, 386, 392, 406, 410, 426, 552, 554, 556, 558, 572, 576, 578, 584, 590, 596, 598, 602, 604, 606, 618, 620, 636, 638, 642, 644, 648, 654, 662, 666, 668, 682, 684, 704, 714, 724, 734, 764, 830, 856, 862, 872, 878, 892, 896, 902, 906, 916, 920, 926, 936, 956, 964, 974, 1634, 1640, 1646, 1654, 1656, 1658, 1664, 1670, 1674, 1682, 1688, 1694, 1698, 1704, 1710, 1712, 1718, 1724, 1766, 1784, 1912, 1918, 1922, 1928, 1934, 1936, 1946, 1952, 1958, 1964, 1970, 2024, 2178, 2184, 2190, 2192, 2198, 2202, 2208, 2214, 2220, 2226, 2232, 2238, 2280, 2294, 2450, 2462, 2466, 2472, 2478, 2492, 2534, 2722, 2724, 2734, 2736, 2746, 2748, 2760, 2774, 2780, 2794, 2804, 3006, 3032, 3044, 3822, 3824, 5736, 5738, 5758, 5762, 5768, 5774, 5782, 5786, 5804, 5822, 5864, 6016, 6022, 6026, 6046, 6076, 6272, 6278, 6282, 6290, 6296, 6302, 6306, 6312, 6318, 6320, 6326, 6332, 6374, 6392, 6544, 6554, 6560, 6566, 6572, 6578, 6632, 6816, 6826, 6828, 6834, 6836, 6846, 6854, 6866, 6872, 6884, 6906, 7088, 7126, 7146, 7904, 7934, 10368, 10374, 10378, 10380, 10388, 10392, 10408, 10414, 10434, 10440, 10454, 10460, 10494, 10646, 10650, 10652, 10666, 10686, 10694, 10700, 10922, 10924, 10944, 10954, 10964, 10974, 11004, 11196, 11204, 11214, 11462, 11468, 11474, 11480, 11492, 11514, 11734, 11754, 12006, 12024, 15554, 15560, 15574, 15594, 15832, 16104, 26754, 26760, 26766, 26774, 26778, 26796, 26814, 26856, 27038, 27068, 27310, 27336, 27370, 27582, 27608, 28398]

representatives of 4-ary noble factions

The top left corner in each 2×2 matrix is a king. The other corners are its equivalents in the other quadrants. (Vertically adjacent quadrants contain relative complements. Horizontally adjacent quadrants differ only in the central vertex.) The two tables below correspond to the image above. The one on the left shows the same numbers as the image. The one on the right shows the smallest Zhegalkin index for each of the 44 factions. (They differ only for the odd factions, i.e. those with green and blue background.) 00 32768128 65534255 32766127 26752104 59520232 38782151 601423 576022 38528150 59774233 27006105 1092042 43688170 54614213 2184685 16326 34400134 63902249 31134121 28384110 61152238 37150145 438217 6802 33448130 64854253 32086125 21768 34944136 63358247 30590119 380814 36576142 61726241 28958113 685626 39624154 58678229 25910101 1146444 44232172 54070211 2130283 00 32768128 65534255 32766127 26752104 59520232 38782151 601423 576022 38528150 48598189 1583061 1092042 43688170 33278129 5101 16326 34400134 60374235 27606107 28384110 61152238 33622131 8543 6802 33448130 43518169 1075041 21768 34944136 40958159 819031 380814 36576142 39326153 655825 685626 39624154 35798139 303011 1146444 44232172 34718135 19507 The black integers are Zhegalkin indices. The gray numbers below are their noble indices, i.e. the positions in the sequence of nobles. The beige numbers in the image are faction sizes, i.e. the number of different permutations of the example shown.

4-ary kings

group under exclusive or

With XOR as a group operation the n-ary noble and royal Boolean functions form a power of the cyclic group C₂.

Python example
The 3-ary nobles form the group C24. The Python operator ^ represents the bitwise XOR. nobles = [0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254] for i, a in enumerate(nobles): for j, b in enumerate(nobles): assert nobles.index(a ^ b) == i ^ j This works for any row of the noble triangle $\Phi$, or from the royal triangle. (But not from the triangle of kings.)

patrons

The XOR of twins is noble, i.e. the XOR of a key and a value in $\Pi _{n}$ is an entry of $\Phi _{n}$.
For a Boolean function this means, that the XOR of its ANF and its truth table is a noble Boolean function, which shall be called its patron.
The patron of a noble is the contradiction.

$\Xi _{n,~k}~~=~~\Pi _{n,~k}\oplus k$

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n 0 0 0 1 0 2 0 2 2 0 14 8 6 8 6 0 14 0 14 8 6 8 6 0 14 row 3 of length 256:     0, 254, 168, 86 ... 168, 86, 0, 254 [0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 128, 126, 40, 214, 72, 182, 224, 30, 0, 254, 168, 86, 200, 54, 96, 158, 96, 158, 200, 54, 168, 86, 0, 254, 224, 30, 72, 182, 40, 214, 128, 126, 0, 254, 168, 86, 200, 54, 96, 158, 128, 126, 40, 214, 72, 182, 224, 30, 224, 30, 72, 182, 40, 214, 128, 126, 96, 158, 200, 54, 168, 86, 0, 254]

For $n\geq 1$ the entries in $\Xi _{n}$ are repetitions of those in $\Phi _{n}$. E.g. $\Xi _{2}$ contains the entries $\{0,6,8,14\}$, each repeated four times.

The set of places where $\Xi _{n}$ has entries $\Phi _{n,~k}$ can be calculated by XORing $\Pi _{n-1,~k}$ with the entries of $\Phi _{n}$.

3-ary Boolean functions with patron 168

$\Pi _{2}\times \Phi _{3}$

Let $\pi =\Pi _{2}$ (length 16), $\phi =\Phi _{3}$ (length 16) and $\xi =\Xi _{3}$ (length 256). Let $I_{k}~~=~~\{i\mid \xi _{i}=\phi _{k}\}$ be the set of places where $\xi$ has the entry $\phi _{k}$. $\pi$ is the left column of the following matrix. $\phi$ 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: $\{\pi _{k}\oplus f\mid f\in \phi \}$ Example:   In which places is $\xi$ equal to $\phi _{10}=168$? $\pi _{10}=2~~~~\implies ~~~~I_{10}~~=~~\{2\oplus f\mid f\in \phi \}~~=~~\{2,28,...,226,252\}$ 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

faction clusters with patron 168
• 2 (dark red), 170 (light yellow) • 84 (light red), 252 (dark yellow) • 42 (dark red), 130 (light yellow) • 124 (light red), 212 (dark yellow) • 28 (dark red), 180 (light yellow) • 52 (dark red), 156 (light yellow) • 74 (light red), 226 (dark yellow) • 98 (light red), 202 (dark yellow)

positions in matrix
Boolean functions with patron 168 in octeract matrix (This is the transpose of Zhegalkin indices with quaestor 3.)