# Linear and noble Boolean functions

| arity | $n$ | 1 | 2 | 3 | 4 | 5 |
| linear | $2^{n+1}$ | 4 | 8 | 16 | 32 | 64 |
| noble | $2^{2^{n-1}}$ | 2 | 4 | 16 | 256 | 65536 |
Among the truth tables for a given arity, the linears and the nobles are important subsets.
Each linear can be assigned a 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 (i.e. the patrons of the linears are a subset of the nobles).

## overview

4-ary :

## 4-ary
### linear to patron (noble)

There are 32 linears, which form 10 factions. (Even and odd faction for each Walsh weight 0...4.) Their patrons are 32 nobles, which also form 10 factions (among the 44 noble factions).
Their juniors are the 3-ary Boolean functions with consul 0. (This is not the case for arities 2 and 3, but probably for all arities ≥ 4.)
list :
| Walsh weight | Walsh index | ¬ | leader | Q | linear | linear | linear | patron (noble) | patron (noble) | patron (noble) |
| Walsh weight | Walsh index | ¬ | leader | Q | truth table | | Ж | truth table | | |
| 0 | 0 | 0 | 0 | 0 | .... .... .... .... | 0 | 0 | .... .... .... .... | 0 | 0 |
| 2 | 3 | 0 | 2 | 0 | .!!. .!!. .!!. .!!. | 26214 | 6 | .... .!!. .!!. .!!. | 26208 | 102 |
| 2 | 5 | 0 | 4 | 0 | .!.! !.!. .!.! !.!. | 23130 | 18 | ...! ..!. .!.! !.!. | 23112 | 90 |
| 2 | 6 | 0 | 6 | 0 | ..!! !!.. ..!! !!.. | 15420 | 20 | ...! .!.. ..!! !!.. | 15400 | 60 |
| 2 | 9 | 0 | 8 | 0 | .!.! .!.! !.!. !.!. | 21930 | 258 | ...! .!.! ..!. !.!. | 21672 | 84 |
| 2 | 10 | 0 | 10 | 0 | ..!! ..!! !!.. !!.. | 13260 | 260 | ...! ..!! .!.. !!.. | 13000 | 50 |
| 2 | 12 | 0 | 12 | 0 | .... !!!! !!!! .... | 4080 | 272 | .... .!!! .!!! .... | 3808 | 14 |
| 4 | 15 | 0 | 14 | 0 | .!!. !..! !..! .!!. | 27030 | 278 | .... ...! ...! .!!. | 26752 | 104 |
| 1 | 1 | 1 | 0 | 1 | !.!. !.!. !.!. !.!. | 21845 | 3 | .!!. !.!. !.!. !.!. | 21846 | 85 |
| 1 | 2 | 1 | 2 | 1 | !!.. !!.. !!.. !!.. | 13107 | 5 | .!!. !!.. !!.. !!.. | 13110 | 51 |
| 1 | 4 | 1 | 4 | 1 | !!!! .... !!!! .... | 3855 | 17 | .!!! !... !!!! .... | 3870 | 15 |
| 3 | 7 | 1 | 6 | 1 | !..! .!!. !..! .!!. | 26985 | 23 | .!!! !!!. !..! .!!. | 27006 | 105 |
| 1 | 8 | 1 | 8 | 1 | !!!! !!!! .... .... | 255 | 257 | .!!! !!!! !... .... | 510 | 1 |
| 3 | 11 | 1 | 10 | 1 | !..! !..! .!!. .!!. | 26265 | 263 | .!!! !..! !!!. .!!. | 26526 | 103 |
| 3 | 13 | 1 | 12 | 1 | !.!. .!.! .!.! !.!. | 23205 | 275 | .!!. !!.! !!.! !.!. | 23478 | 91 |
| 3 | 14 | 1 | 14 | 1 | !!.. ..!! ..!! !!.. | 15555 | 277 | .!!. !.!! !.!! !!.. | 15830 | 61 |
| 1 | 1 | 0 | 0 | 2 | .!.! .!.! .!.! .!.! | 43690 | 2 | ...! .!.! .!.! .!.! | 43688 | 170 |
| 1 | 2 | 0 | 2 | 2 | ..!! ..!! ..!! ..!! | 52428 | 4 | ...! ..!! ..!! ..!! | 52424 | 204 |
| 1 | 4 | 0 | 4 | 2 | .... !!!! .... !!!! | 61680 | 16 | .... .!!! .... !!!! | 61664 | 240 |
| 3 | 7 | 0 | 6 | 2 | .!!. !..! .!!. !..! | 38550 | 22 | .... ...! .!!. !..! | 38528 | 150 |
| 1 | 8 | 0 | 8 | 2 | .... .... !!!! !!!! | 65280 | 256 | .... .... .!!! !!!! | 65024 | 254 |
| 3 | 11 | 0 | 10 | 2 | .!!. .!!. !..! !..! | 39270 | 262 | .... .!!. ...! !..! | 39008 | 152 |
| 3 | 13 | 0 | 12 | 2 | .!.! !.!. !.!. .!.! | 42330 | 274 | ...! ..!. ..!. .!.! | 42056 | 164 |
| 3 | 14 | 0 | 14 | 2 | ..!! !!.. !!.. ..!! | 49980 | 276 | ...! .!.. .!.. ..!! | 49704 | 194 |
| 0 | 0 | 1 | 0 | 3 | !!!! !!!! !!!! !!!! | 65535 | 1 | .!!! !!!! !!!! !!!! | 65534 | 255 |
| 2 | 3 | 1 | 2 | 3 | !..! !..! !..! !..! | 39321 | 7 | .!!! !..! !..! !..! | 39326 | 153 |
| 2 | 5 | 1 | 4 | 3 | !.!. .!.! !.!. .!.! | 42405 | 19 | .!!. !!.! !.!. .!.! | 42422 | 165 |
| 2 | 6 | 1 | 6 | 3 | !!.. ..!! !!.. ..!! | 50115 | 21 | .!!. !.!! !!.. ..!! | 50134 | 195 |
| 2 | 9 | 1 | 8 | 3 | !.!. !.!. .!.! .!.! | 43605 | 259 | .!!. !.!. !!.! .!.! | 43862 | 171 |
| 2 | 10 | 1 | 10 | 3 | !!.. !!.. ..!! ..!! | 52275 | 261 | .!!. !!.. !.!! ..!! | 52534 | 205 |
| 2 | 12 | 1 | 12 | 3 | !!!! .... .... !!!! | 61455 | 273 | .!!! !... !... !!!! | 61726 | 241 |
| 4 | 15 | 1 | 14 | 3 | !..! .!!. .!!. !..! | 38505 | 279 | .!!! !!!. !!!. !..! | 38782 | 151 |
details for each faction :
| Walsh weight | Walsh index | faction size | Walsh | Walsh | patron | patron | twin | ¬ Walsh | ¬ Walsh | patron | patron | twin |
| Walsh weight | Walsh index | faction size | Q | | | | | Q | | | | |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 3 | 65535 | 65534 | 255 | 1 |
| 1 | 1 | 4 | 2 | 43690 | 43688 | 170 | 2 | 1 | 21845 | 21846 | 85 | 3 |
| 2 | 3 | 6 | 0 | 26214 | 26208 | 102 | 6 | 3 | 39321 | 39326 | 153 | 7 |
| 3 | 7 | 4 | 2 | 38550 | 38528 | 150 | 22 | 1 | 26985 | 27006 | 105 | 23 |
| 4 | 15 | 1 | 0 | 27030 | 26752 | 104 | 278 | 3 | 38505 | 38782 | 151 | 279 |
3-ary noble indices   (juniors) :
The factions are shown in ten different colors, which are variants of the quadrant colors.  
(There are three shades of red and green, and two shades of blue and yellow.) :
| truth tables | Zhegalkin indices |
The following images show that pairs of complementary factions are in the same principality.
quadrants 0 and 3 :
| king | 0 | 3808 | 26752 |
| king index | 0 | 14 | 104 |
quadrants 1 and 2 :
| king | 5760 | 10920 |
| king index | 22 | 42 |

### noble to prefect (linear)

There are 256 nobles. A prefect is one of the 32 linears. Every linear is the prefect of 8 nobles.
list   (32 × 8 matrix) :
| Walsh weight | Walsh index | ¬ | leader | Q | prefect (linear) | prefect (linear) | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles | nobles |
| Walsh weight | Walsh index | ¬ | leader | Q | | Ж | | | | | | | | | | | | | | | | | | | | | | | | |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | (0) | 854 | 3 | (110) | 1334 | 5 | (110) | 1632 | 6 | (6) | 4382 | 17 | (110) | 4680 | 18 | (6) | 5160 | 20 | (6) | 6014 | 23 | (104) |
| 2 | 3 | 0 | 2 | 0 | 26214 | 6 | 26208 | 102 | (14) | 25910 | 101 | (26) | 25430 | 99 | (26) | 24576 | 96 | (8) | 30590 | 119 | (8) | 29736 | 116 | (26) | 29256 | 114 | (26) | 28958 | 113 | (14) |
| 2 | 5 | 0 | 4 | 0 | 23130 | 18 | 23112 | 90 | (14) | 22814 | 89 | (26) | 24446 | 95 | (8) | 23592 | 92 | (26) | 19286 | 75 | (26) | 18432 | 72 | (8) | 20064 | 78 | (26) | 19766 | 77 | (14) |
| 2 | 6 | 0 | 6 | 0 | 15420 | 20 | 15400 | 60 | (14) | 16254 | 63 | (8) | 14622 | 57 | (26) | 14920 | 58 | (26) | 11574 | 45 | (26) | 11872 | 46 | (26) | 10240 | 40 | (8) | 11094 | 43 | (14) |
| 2 | 9 | 0 | 8 | 0 | 21930 | 258 | 21672 | 84 | (14) | 22526 | 87 | (8) | 20894 | 81 | (26) | 21192 | 82 | (26) | 17846 | 69 | (26) | 18144 | 70 | (26) | 16512 | 64 | (8) | 17366 | 67 | (14) |
| 2 | 10 | 0 | 10 | 0 | 13260 | 260 | 13000 | 50 | (14) | 12702 | 49 | (26) | 14334 | 55 | (8) | 13480 | 52 | (26) | 9174 | 35 | (26) | 8320 | 32 | (8) | 9952 | 38 | (26) | 9654 | 37 | (14) |
| 2 | 12 | 0 | 12 | 0 | 4080 | 272 | 3808 | 14 | (14) | 3510 | 13 | (26) | 3030 | 11 | (26) | 2176 | 8 | (8) | 8190 | 31 | (8) | 7336 | 28 | (26) | 6856 | 26 | (26) | 6558 | 25 | (14) |
| 4 | 15 | 0 | 14 | 0 | 27030 | 278 | 26752 | 104 | (104) | 27606 | 107 | (6) | 28086 | 109 | (6) | 28384 | 110 | (110) | 31134 | 121 | (6) | 31432 | 122 | (110) | 31912 | 124 | (110) | 32766 | 127 | (0) |
| 1 | 1 | 1 | 0 | 1 | 21845 | 3 | 21846 | 85 | (42) | 22016 | 86 | (2) | 20576 | 80 | (2) | 21302 | 83 | (44) | 17480 | 68 | (2) | 18206 | 71 | (44) | 16766 | 65 | (44) | 16936 | 66 | (22) |
| 1 | 2 | 1 | 2 | 1 | 13107 | 5 | 13110 | 51 | (42) | 12384 | 48 | (2) | 13824 | 54 | (2) | 13654 | 53 | (44) | 8744 | 34 | (2) | 8574 | 33 | (44) | 10014 | 39 | (44) | 9288 | 36 | (22) |
| 1 | 4 | 1 | 4 | 1 | 3855 | 17 | 3870 | 15 | (42) | 3144 | 12 | (2) | 2600 | 10 | (2) | 2430 | 9 | (44) | 7680 | 30 | (2) | 7510 | 29 | (44) | 6966 | 27 | (44) | 6240 | 24 | (22) |
| 3 | 7 | 1 | 6 | 1 | 26985 | 23 | 27006 | 105 | (22) | 27176 | 106 | (44) | 27720 | 108 | (44) | 28446 | 111 | (2) | 30816 | 120 | (44) | 31542 | 123 | (2) | 32086 | 125 | (2) | 32256 | 126 | (42) |
| 1 | 8 | 1 | 8 | 1 | 255 | 257 | 510 | 1 | (42) | 680 | 2 | (2) | 1224 | 4 | (2) | 1950 | 7 | (44) | 4320 | 16 | (2) | 5046 | 19 | (44) | 5590 | 21 | (44) | 5760 | 22 | (22) |
| 3 | 11 | 1 | 10 | 1 | 26265 | 263 | 26526 | 103 | (22) | 25800 | 100 | (44) | 25256 | 98 | (44) | 25086 | 97 | (2) | 30336 | 118 | (44) | 30166 | 117 | (2) | 29622 | 115 | (2) | 28896 | 112 | (42) |
| 3 | 13 | 1 | 12 | 1 | 23205 | 275 | 23478 | 91 | (22) | 22752 | 88 | (44) | 24192 | 94 | (44) | 24022 | 93 | (2) | 19112 | 74 | (44) | 18942 | 73 | (2) | 20382 | 79 | (2) | 19656 | 76 | (42) |
| 3 | 14 | 1 | 14 | 1 | 15555 | 277 | 15830 | 61 | (22) | 16000 | 62 | (44) | 14560 | 56 | (44) | 15286 | 59 | (2) | 11464 | 44 | (44) | 12190 | 47 | (2) | 10750 | 41 | (2) | 10920 | 42 | (42) |
| 1 | 1 | 0 | 0 | 2 | 43690 | 2 | 43688 | 170 | (42) | 43518 | 169 | (2) | 44958 | 175 | (2) | 44232 | 172 | (44) | 48054 | 187 | (2) | 47328 | 184 | (44) | 48768 | 190 | (44) | 48598 | 189 | (22) |
| 1 | 2 | 0 | 2 | 2 | 52428 | 4 | 52424 | 204 | (42) | 53150 | 207 | (2) | 51710 | 201 | (2) | 51880 | 202 | (44) | 56790 | 221 | (2) | 56960 | 222 | (44) | 55520 | 216 | (44) | 56246 | 219 | (22) |
| 1 | 4 | 0 | 4 | 2 | 61680 | 16 | 61664 | 240 | (42) | 62390 | 243 | (2) | 62934 | 245 | (2) | 63104 | 246 | (44) | 57854 | 225 | (2) | 58024 | 226 | (44) | 58568 | 228 | (44) | 59294 | 231 | (22) |
| 3 | 7 | 0 | 6 | 2 | 38550 | 22 | 38528 | 150 | (22) | 38358 | 149 | (44) | 37814 | 147 | (44) | 37088 | 144 | (2) | 34718 | 135 | (44) | 33992 | 132 | (2) | 33448 | 130 | (2) | 33278 | 129 | (42) |
| 1 | 8 | 0 | 8 | 2 | 65280 | 256 | 65024 | 254 | (42) | 64854 | 253 | (2) | 64310 | 251 | (2) | 63584 | 248 | (44) | 61214 | 239 | (2) | 60488 | 236 | (44) | 59944 | 234 | (44) | 59774 | 233 | (22) |
| 3 | 11 | 0 | 10 | 2 | 39270 | 262 | 39008 | 152 | (22) | 39734 | 155 | (44) | 40278 | 157 | (44) | 40448 | 158 | (2) | 35198 | 137 | (44) | 35368 | 138 | (2) | 35912 | 140 | (2) | 36638 | 143 | (42) |
| 3 | 13 | 0 | 12 | 2 | 42330 | 274 | 42056 | 164 | (22) | 42782 | 167 | (44) | 41342 | 161 | (44) | 41512 | 162 | (2) | 46422 | 181 | (44) | 46592 | 182 | (2) | 45152 | 176 | (2) | 45878 | 179 | (42) |
| 3 | 14 | 0 | 14 | 2 | 49980 | 276 | 49704 | 194 | (22) | 49534 | 193 | (44) | 50974 | 199 | (44) | 50248 | 196 | (2) | 54070 | 211 | (44) | 53344 | 208 | (2) | 54784 | 214 | (2) | 54614 | 213 | (42) |
| 0 | 0 | 1 | 0 | 3 | 65535 | 1 | 65534 | 255 | (0) | 64680 | 252 | (110) | 64200 | 250 | (110) | 63902 | 249 | (6) | 61152 | 238 | (110) | 60854 | 237 | (6) | 60374 | 235 | (6) | 59520 | 232 | (104) |
| 2 | 3 | 1 | 2 | 3 | 39321 | 7 | 39326 | 153 | (14) | 39624 | 154 | (26) | 40104 | 156 | (26) | 40958 | 159 | (8) | 34944 | 136 | (8) | 35798 | 139 | (26) | 36278 | 141 | (26) | 36576 | 142 | (14) |
| 2 | 5 | 1 | 4 | 3 | 42405 | 19 | 42422 | 165 | (14) | 42720 | 166 | (26) | 41088 | 160 | (8) | 41942 | 163 | (26) | 46248 | 180 | (26) | 47102 | 183 | (8) | 45470 | 177 | (26) | 45768 | 178 | (14) |
| 2 | 6 | 1 | 6 | 3 | 50115 | 21 | 50134 | 195 | (14) | 49280 | 192 | (8) | 50912 | 198 | (26) | 50614 | 197 | (26) | 53960 | 210 | (26) | 53662 | 209 | (26) | 55294 | 215 | (8) | 54440 | 212 | (14) |
| 2 | 9 | 1 | 8 | 3 | 43605 | 259 | 43862 | 171 | (14) | 43008 | 168 | (8) | 44640 | 174 | (26) | 44342 | 173 | (26) | 47688 | 186 | (26) | 47390 | 185 | (26) | 49022 | 191 | (8) | 48168 | 188 | (14) |
| 2 | 10 | 1 | 10 | 3 | 52275 | 261 | 52534 | 205 | (14) | 52832 | 206 | (26) | 51200 | 200 | (8) | 52054 | 203 | (26) | 56360 | 220 | (26) | 57214 | 223 | (8) | 55582 | 217 | (26) | 55880 | 218 | (14) |
| 2 | 12 | 1 | 12 | 3 | 61455 | 273 | 61726 | 241 | (14) | 62024 | 242 | (26) | 62504 | 244 | (26) | 63358 | 247 | (8) | 57344 | 224 | (8) | 58198 | 227 | (26) | 58678 | 229 | (26) | 58976 | 230 | (14) |
| 4 | 15 | 1 | 14 | 3 | 38505 | 279 | 38782 | 151 | (104) | 37928 | 148 | (6) | 37448 | 146 | (6) | 37150 | 145 | (110) | 34400 | 134 | (6) | 34102 | 133 | (110) | 33622 | 131 | (110) | 32768 | 128 | (0) |
3-ary subprefects   (16 × 16 matrices) :
| | leader | signed Walsh indices |
| 0 | 0 | 0  ¬1  1  ¬0 |
| 0 | 8 | 9  ¬8  8  ¬9 |
| 4 | 4 | 5  ¬4  4  ¬5 |
| 4 | 12 | 12  ¬13  13  ¬12 |
| 2 | 2 | 3  ¬2  2  ¬3 |
| 2 | 10 | 10  ¬11  11  ¬10 |
| 6 | 6 | 6  ¬7  7  ¬6 |
| 6 | 14 | 15  ¬14  14  ¬15 |