# 4-ary Boolean functions

There are $2^{2^{4}}$ = 65536 4-ary Boolean functions, which correspond to 16-bit binary strings.

## Big equivalence classes (bec)

       Compare: Big equivalence classes of 3-ary Boolean functions
When binarily colored tesseracts can be transformed into each other by actions in the tesseract's symmetry group C$_4$, they are essentially the same. The corresponding Boolean functions are often called equivalent and belong to the same big equivalence class (bec). The 65536 functions belong to 1 + 1 + 4 + 6 + 19 + 27 + 50 + 56 + 74 + 56 + 50 + 27 + 19 + 6 + 4 + 1 + 1 = 402 becs.
The following sortable table (in the collapsed box) shows the 402 becs with some key features. The default order is that of the numbers in column N. Sorting by # sorts also by N. The numbers in column weight are the hamming weights of the functions, i.e. the number of ones in the binary strings. Column sec tells the number of secs in the bec. Column f tells the number of functions in the bec.
The function with the smallest numerical value is shown in column F as an example. It's numerical value is shown in column N. becs with an entry in column sona are related to subgroups of nimber addition. The entry in column wec denotes the Walsh equivalence class the bec belongs to. The nonlinearity of all functions in the bec is shown in column nonlin. Functions with nonlinearity 0 are linear, those with nonlinearity 6 are bent. To sort by more than one column you can hold the shift-key and than click the other sort buttons. *
Table of the 402 becs :
| # | weight | sec | f | F | N | sona | wec | nonlin |
| 0 | 0 | 1 | 1 | .... .... .... .... | 0 | | E0 | 0 |
| 1 | 1 | 1 | 16 | !... .... .... .... | 1 | N(0) | O | 1 |
| 2 | 2 | 4 | 32 | !!.. .... .... .... | 3 | N(1,1) | E1 | 2 |
| 3 | 2 | 6 | 48 | .!!. .... .... .... | 6 | N(1,2) | E2 | 2 |
| 4 | 3 | 6 | 96 | !!!. .... .... .... | 7 | | O | 3 |
| 5 | 4 | 6 | 24 | !!!! .... .... .... | 15 | N(2,1) | E0 | 4 |
| 6 | 3 | 4 | 64 | .!!. !... .... .... | 22 | | O | 3 |
| 7 | 4 | 4 | 64 | !!!. !... .... .... | 23 | | E3 | 4 |
| 8 | 2 | 4 | 32 | ...! !... .... .... | 24 | N(1,3) | E3 | 2 |
| 9 | 3 | 12 | 192 | !..! !... .... .... | 25 | | O | 3 |
| 10 | 4 | 12 | 192 | !!.! !... .... .... | 27 | | E2 | 4 |
| 11 | 4 | 12 | 192 | .!!! !... .... .... | 30 | | E1 | 4 |
| 12 | 5 | 12 | 192 | !!!! !... .... .... | 31 | | O | 3 |
| 13 | 4 | 12 | 48 | ..!! !!.. .... .... | 60 | N(2,2) | E0 | 4 |
| 14 | 5 | 12 | 192 | !.!! !!.. .... .... | 61 | | O | 3 |
| 15 | 6 | 12 | 96 | !!!! !!.. .... .... | 63 | | E1 | 2 |
| 16 | 4 | 4 | 16 | !..! .!!. .... .... | 105 | N(2,3) | E0 | 4 |
| 17 | 5 | 4 | 64 | !!.! .!!. .... .... | 107 | | O | 3 |
| 18 | 6 | 12 | 96 | !!!! .!!. .... .... | 111 | | E2 | 2 |
| 19 | 6 | 4 | 32 | .!!! !!!. .... .... | 126 | | E3 | 2 |
| 20 | 7 | 4 | 64 | !!!! !!!. .... .... | 127 | | O | 1 |
| 21 | 8 | 4 | 8 | !!!! !!!! .... .... | 255 | N(3,1) | E0 | 0 |
| 22 | 4 | 1 | 16 | .!!. !... !... .... | 278 | | E4 | 4 |
| 23 | 5 | 1 | 16 | !!!. !... !... .... | 279 | | O | 5 |
| 24 | 3 | 6 | 96 | ...! !... !... .... | 280 | | O | 3 |
| 25 | 4 | 6 | 96 | !..! !... !... .... | 281 | | E4 | 4 |
| 26 | 4 | 12 | 192 | .!.! !... !... .... | 282 | | E3 | 4 |
| 27 | 5 | 12 | 192 | !!.! !... !... .... | 283 | | O | 5 |
| 28 | 5 | 6 | 96 | .!!! !... !... .... | 286 | | O | 5 |
| 29 | 6 | 6 | 96 | !!!! !... !... .... | 287 | | E2 | 4 |
| 30 | 4 | 12 | 192 | ..!! .!.. !... .... | 300 | | E2 | 4 |
| 31 | 5 | 24 | 384 | !.!! .!.. !... .... | 301 | | O | 5 |
| 32 | 6 | 12 | 192 | !!!! .!.. !... .... | 303 | | E3 | 4 |
| 33 | 5 | 12 | 192 | ..!! !!.. !... .... | 316 | | O | 3 |
| 34 | 6 | 12 | 192 | !.!! !!.. !... .... | 317 | | E1 | 4 |
| 35 | 6 | 12 | 192 | .!!! !!.. !... .... | 318 | | E2 | 4 |
| 36 | 7 | 12 | 192 | !!!! !!.. !... .... | 319 | | O | 3 |
| 37 | 4 | 4 | 64 | ...! .!!. !... .... | 360 | | E1 | 4 |
| 38 | 5 | 4 | 64 | !..! .!!. !... .... | 361 | | O | 3 |
| 39 | 5 | 12 | 192 | .!.! .!!. !... .... | 362 | | O | 5 |
| 40 | 6 | 12 | 192 | !!.! .!!. !... .... | 363 | | E2 | 4 |
| 41 | 6 | 12 | 192 | .!!! .!!. !... .... | 366 | | E3 | 4 |
| 42 | 7 | 12 | 192 | !!!! .!!. !... .... | 367 | | O | 3 |
| 43 | 7 | 4 | 64 | .!!! !!!. !... .... | 382 | | O | 3 |
| 44 | 8 | 4 | 64 | !!!! !!!. !... .... | 383 | | E4 | 2 |
| 45 | 2 | 1 | 8 | .... ...! !... .... | 384 | N(1,4) | E4 | 2 |
| 46 | 3 | 4 | 64 | !... ...! !... .... | 385 | | O | 3 |
| 47 | 3 | 3 | 48 | .!.. ...! !... .... | 386 | | O | 3 |
| 48 | 4 | 12 | 192 | !!.. ...! !... .... | 387 | | E3 | 4 |
| 49 | 4 | 6 | 96 | .!!. ...! !... .... | 390 | | E2 | 4 |
| 50 | 5 | 12 | 192 | !!!. ...! !... .... | 391 | | O | 5 |
| 51 | 4 | 6 | 96 | !..! ...! !... .... | 393 | | E2 | 4 |
| 52 | 5 | 12 | 192 | !!.! ...! !... .... | 395 | | O | 5 |
| 53 | 6 | 6 | 96 | !!!! ...! !... .... | 399 | | E4 | 4 |
| 54 | 5 | 4 | 64 | .!!. !..! !... .... | 406 | | O | 3 |
| 55 | 6 | 4 | 64 | !!!. !..! !... .... | 407 | | E1 | 4 |
| 56 | 4 | 12 | 192 | ...! !..! !... .... | 408 | | E1 | 4 |
| 57 | 5 | 12 | 192 | !..! !..! !... .... | 409 | | O | 3 |
| 58 | 5 | 24 | 384 | .!.! !..! !... .... | 410 | | O | 5 |
| 59 | 6 | 24 | 384 | !!.! !..! !... .... | 411 | | E2 | 4 |
| 60 | 6 | 12 | 192 | .!!! !..! !... .... | 414 | | E3 | 4 |
| 61 | 7 | 12 | 192 | !!!! !..! !... .... | 415 | | O | 3 |
| 62 | 4 | 6 | 96 | ...! .!.! !... .... | 424 | | E2 | 4 |
| 63 | 5 | 12 | 192 | !..! .!.! !... .... | 425 | | O | 5 |
| 64 | 5 | 6 | 96 | .!.! .!.! !... .... | 426 | | O | 3 |
| 65 | 6 | 12 | 192 | !!.! .!.! !... .... | 427 | | E1 | 4 |
| 66 | 5 | 24 | 384 | ..!! .!.! !... .... | 428 | | O | 5 |
| 67 | 6 | 24 | 384 | !.!! .!.! !... .... | 429 | | E3 | 4 |
| 68 | 6 | 24 | 384 | .!!! .!.! !... .... | 430 | | E2 | 4 |
| 69 | 7 | 24 | 384 | !!!! .!.! !... .... | 431 | | O | 3 |
| 70 | 6 | 12 | 192 | ..!! !!.! !... .... | 444 | | E4 | 4 |
| 71 | 7 | 12 | 192 | !.!! !!.! !... .... | 445 | | O | 3 |
| 72 | 7 | 12 | 192 | .!!! !!.! !... .... | 446 | | O | 3 |
| 73 | 8 | 12 | 192 | !!!! !!.! !... .... | 447 | | E3 | 2 |
| 74 | 5 | 4 | 64 | ...! .!!! !... .... | 488 | | O | 5 |
| 75 | 6 | 4 | 64 | !..! .!!! !... .... | 489 | | E4 | 4 |
| 76 | 6 | 12 | 192 | .!.! .!!! !... .... | 490 | | E3 | 4 |
| 77 | 7 | 12 | 192 | !!.! .!!! !... .... | 491 | | O | 3 |
| 78 | 7 | 12 | 192 | .!!! .!!! !... .... | 494 | | O | 3 |
| 79 | 8 | 12 | 192 | !!!! .!!! !... .... | 495 | | E2 | 2 |
| 80 | 8 | 4 | 64 | .!!! !!!! !... .... | 510 | | E1 | 2 |
| 81 | 9 | 4 | 64 | !!!! !!!! !... .... | 511 | | O | 1 |
| 82 | 6 | 4 | 32 | ..!! !!.. !!.. .... | 828 | | E1 | 2 |
| 83 | 7 | 4 | 64 | !.!! !!.. !!.. .... | 829 | | O | 3 |
| 84 | 8 | 4 | 32 | !!!! !!.. !!.. .... | 831 | | E0 | 4 |
| 85 | 6 | 3 | 48 | .!!. !.!. !!.. .... | 854 | | E0 | 6 |
| 86 | 7 | 3 | 48 | !!!. !.!. !!.. .... | 855 | | O | 5 |
| 87 | 5 | 12 | 192 | ...! !.!. !!.. .... | 856 | | O | 5 |
| 88 | 6 | 12 | 192 | !..! !.!. !!.. .... | 857 | | E0 | 6 |
| 89 | 6 | 24 | 384 | .!.! !.!. !!.. .... | 858 | | E1 | 4 |
| 90 | 7 | 24 | 384 | !!.! !.!. !!.. .... | 859 | | O | 5 |
| 91 | 7 | 12 | 192 | .!!! !.!. !!.. .... | 862 | | O | 5 |
| 92 | 8 | 12 | 192 | !!!! !.!. !!.. .... | 863 | | E2 | 4 |
| 93 | 5 | 12 | 192 | ...! .!!. !!.. .... | 872 | | O | 5 |
| 94 | 6 | 12 | 192 | !..! .!!. !!.. .... | 873 | | E1 | 4 |
| 95 | 6 | 12 | 192 | .!.! .!!. !!.. .... | 874 | | E0 | 6 |
| 96 | 7 | 12 | 192 | !!.! .!!. !!.. .... | 875 | | O | 5 |
| 97 | 6 | 24 | 384 | ..!! .!!. !!.. .... | 876 | | E2 | 4 |
| 98 | 7 | 24 | 384 | !.!! .!!. !!.. .... | 877 | | O | 5 |
| 99 | 7 | 24 | 384 | .!!! .!!. !!.. .... | 878 | | O | 5 |
| 100 | 8 | 24 | 384 | !!!! .!!. !!.. .... | 879 | | E1 | 4 |
| 101 | 7 | 12 | 192 | ..!! !!!. !!.. .... | 892 | | O | 3 |
| 102 | 8 | 12 | 192 | !.!! !!!. !!.. .... | 893 | | E3 | 4 |
| 103 | 8 | 12 | 192 | .!!! !!!. !!.. .... | 894 | | E2 | 4 |
| 104 | 9 | 12 | 192 | !!!! !!!. !!.. .... | 895 | | O | 3 |
| 105 | 4 | 4 | 16 | .... ..!! !!.. .... | 960 | N(2,4) | E0 | 4 |
| 106 | 5 | 12 | 192 | !... ..!! !!.. .... | 961 | | O | 3 |
| 107 | 6 | 12 | 96 | !!.. ..!! !!.. .... | 963 | | E1 | 2 |
| 108 | 6 | 12 | 192 | !.!. ..!! !!.. .... | 965 | | E1 | 4 |
| 109 | 6 | 12 | 192 | .!!. ..!! !!.. .... | 966 | | E2 | 4 |
| 110 | 7 | 24 | 384 | !!!. ..!! !!.. .... | 967 | | O | 3 |
| 111 | 8 | 12 | 96 | !!!! ..!! !!.. .... | 975 | | E0 | 4 |
| 112 | 6 | 12 | 192 | ..!. !.!! !!.. .... | 980 | | E2 | 4 |
| 113 | 7 | 12 | 192 | !.!. !.!! !!.. .... | 981 | | O | 5 |
| 114 | 7 | 12 | 192 | .!!. !.!! !!.. .... | 982 | | O | 5 |
| 115 | 8 | 12 | 192 | !!!. !.!! !!.. .... | 983 | | E3 | 4 |
| 116 | 6 | 12 | 192 | ...! !.!! !!.. .... | 984 | | E3 | 4 |
| 117 | 7 | 24 | 384 | !..! !.!! !!.. .... | 985 | | O | 5 |
| 118 | 8 | 12 | 192 | !!.! !.!! !!.. .... | 987 | | E2 | 4 |
| 119 | 7 | 24 | 384 | ..!! !.!! !!.. .... | 988 | | O | 3 |
| 120 | 8 | 24 | 384 | !.!! !.!! !!.. .... | 989 | | E2 | 4 |
| 121 | 8 | 24 | 384 | .!!! !.!! !!.. .... | 990 | | E1 | 4 |
| 122 | 9 | 24 | 384 | !!!! !.!! !!.. .... | 991 | | O | 3 |
| 123 | 8 | 12 | 96 | ..!! !!!! !!.. .... | 1020 | | E0 | 4 |
| 124 | 9 | 12 | 192 | !.!! !!!! !!.. .... | 1021 | | O | 3 |
| 125 | 10 | 12 | 96 | !!!! !!!! !!.. .... | 1023 | | E1 | 2 |
| 126 | 4 | 3 | 12 | .... .!!. .!!. .... | 1632 | N(2,5) | E0 | 4 |
| 127 | 5 | 3 | 48 | !... .!!. .!!. .... | 1633 | | O | 3 |
| 128 | 5 | 6 | 96 | .!.. .!!. .!!. .... | 1634 | | O | 3 |
| 129 | 6 | 12 | 192 | !!.. .!!. .!!. .... | 1635 | | E1 | 4 |
| 130 | 6 | 6 | 48 | .!!. .!!. .!!. .... | 1638 | | E2 | 2 |
| 131 | 7 | 6 | 96 | !!!. .!!. .!!. .... | 1639 | | O | 3 |
| 132 | 6 | 6 | 48 | !..! .!!. .!!. .... | 1641 | | E2 | 2 |
| 133 | 7 | 6 | 96 | !!.! .!!. .!!. .... | 1643 | | O | 3 |
| 134 | 8 | 6 | 48 | !!!! .!!. .!!. .... | 1647 | | E0 | 4 |
| 135 | 6 | 12 | 192 | .!.. !!!. .!!. .... | 1650 | | E2 | 4 |
| 136 | 7 | 12 | 192 | !!.. !!!. .!!. .... | 1651 | | O | 5 |
| 137 | 7 | 12 | 192 | .!!. !!!. .!!. .... | 1654 | | O | 3 |
| 138 | 8 | 12 | 192 | !!!. !!!. .!!. .... | 1655 | | E3 | 4 |
| 139 | 6 | 12 | 192 | ...! !!!. .!!. .... | 1656 | | E3 | 4 |
| 140 | 7 | 12 | 192 | !..! !!!. .!!. .... | 1657 | | O | 3 |
| 141 | 7 | 24 | 384 | .!.! !!!. .!!. .... | 1658 | | O | 5 |
| 142 | 8 | 24 | 384 | !!.! !!!. .!!. .... | 1659 | | E2 | 4 |
| 143 | 8 | 12 | 192 | .!!! !!!. .!!. .... | 1662 | | E1 | 4 |
| 144 | 9 | 12 | 192 | !!!! !!!. .!!. .... | 1663 | | O | 3 |
| 145 | 4 | 6 | 24 | .... !..! .!!. .... | 1680 | N(2,6) | E0 | 4 |
| 146 | 5 | 12 | 192 | !... !..! .!!. .... | 1681 | | O | 3 |
| 147 | 6 | 12 | 192 | !!.. !..! .!!. .... | 1683 | | E1 | 4 |
| 148 | 6 | 12 | 96 | .!!. !..! .!!. .... | 1686 | | E2 | 2 |
| 149 | 7 | 12 | 192 | !!!. !..! .!!. .... | 1687 | | O | 3 |
| 150 | 8 | 6 | 48 | !!!! !..! .!!. .... | 1695 | | E0 | 4 |
| 151 | 5 | 6 | 96 | .... !!.! .!!. .... | 1712 | | O | 3 |
| 152 | 6 | 24 | 384 | !... !!.! .!!. .... | 1713 | | E2 | 4 |
| 153 | 6 | 12 | 192 | .!.. !!.! .!!. .... | 1714 | | E1 | 4 |
| 154 | 7 | 24 | 384 | !!.. !!.! .!!. .... | 1715 | | O | 5 |
| 155 | 6 | 12 | 192 | ..!. !!.! .!!. .... | 1716 | | E3 | 4 |
| 156 | 7 | 24 | 384 | !.!. !!.! .!!. .... | 1717 | | O | 5 |
| 157 | 7 | 12 | 192 | .!!. !!.! .!!. .... | 1718 | | O | 3 |
| 158 | 8 | 24 | 384 | !!!. !!.! .!!. .... | 1719 | | E2 | 4 |
| 159 | 7 | 12 | 192 | !..! !!.! .!!. .... | 1721 | | O | 3 |
| 160 | 8 | 12 | 192 | !!.! !!.! .!!. .... | 1723 | | E3 | 4 |
| 161 | 8 | 12 | 192 | !.!! !!.! .!!. .... | 1725 | | E1 | 4 |
| 162 | 9 | 12 | 192 | !!!! !!.! .!!. .... | 1727 | | O | 3 |
| 163 | 6 | 6 | 48 | .... !!!! .!!. .... | 1776 | | E2 | 2 |
| 164 | 7 | 12 | 192 | !... !!!! .!!. .... | 1777 | | O | 3 |
| 165 | 7 | 12 | 192 | .!.. !!!! .!!. .... | 1778 | | O | 3 |
| 166 | 8 | 24 | 384 | !!.. !!!! .!!. .... | 1779 | | E1 | 4 |
| 167 | 8 | 12 | 96 | .!!. !!!! .!!. .... | 1782 | | E0 | 4 |
| 168 | 9 | 12 | 192 | !!!. !!!! .!!. .... | 1783 | | O | 3 |
| 169 | 8 | 12 | 96 | !..! !!!! .!!. .... | 1785 | | E0 | 4 |
| 170 | 9 | 12 | 192 | !!.! !!!! .!!. .... | 1787 | | O | 3 |
| 171 | 10 | 12 | 96 | !!!! !!!! .!!. .... | 1791 | | E2 | 2 |
| 172 | 8 | 3 | 48 | .!!. !!!. !!!. .... | 1910 | | E4 | 4 |
| 173 | 9 | 3 | 48 | !!!. !!!. !!!. .... | 1911 | | O | 5 |
| 174 | 7 | 6 | 96 | ...! !!!. !!!. .... | 1912 | | O | 3 |
| 175 | 8 | 6 | 96 | !..! !!!. !!!. .... | 1913 | | E4 | 4 |
| 176 | 8 | 12 | 192 | .!.! !!!. !!!. .... | 1914 | | E3 | 4 |
| 177 | 9 | 12 | 192 | !!.! !!!. !!!. .... | 1915 | | O | 5 |
| 178 | 9 | 6 | 96 | .!!! !!!. !!!. .... | 1918 | | O | 5 |
| 179 | 10 | 6 | 96 | !!!! !!!. !!!. .... | 1919 | | E2 | 4 |
| 180 | 6 | 12 | 96 | .... !!.! !!!. .... | 1968 | | E3 | 2 |
| 181 | 7 | 24 | 384 | !... !!.! !!!. .... | 1969 | | O | 3 |
| 182 | 8 | 12 | 192 | !!.. !!.! !!!. .... | 1971 | | E2 | 4 |
| 183 | 7 | 24 | 384 | ..!. !!.! !!!. .... | 1972 | | O | 3 |
| 184 | 8 | 24 | 384 | !.!. !!.! !!!. .... | 1973 | | E4 | 4 |
| 185 | 8 | 24 | 384 | .!!. !!.! !!!. .... | 1974 | | E3 | 4 |
| 186 | 9 | 24 | 384 | !!!. !!.! !!!. .... | 1975 | | O | 5 |
| 187 | 8 | 12 | 192 | ..!! !!.! !!!. .... | 1980 | | E2 | 4 |
| 188 | 9 | 24 | 384 | !.!! !!.! !!!. .... | 1981 | | O | 5 |
| 189 | 10 | 12 | 192 | !!!! !!.! !!!. .... | 1983 | | E3 | 4 |
| 190 | 6 | 6 | 48 | .... .!!! !!!. .... | 2016 | | E4 | 2 |
| 191 | 7 | 12 | 192 | !... .!!! !!!. .... | 2017 | | O | 3 |
| 192 | 7 | 12 | 192 | .!.. .!!! !!!. .... | 2018 | | O | 3 |
| 193 | 8 | 24 | 384 | !!.. .!!! !!!. .... | 2019 | | E3 | 4 |
| 194 | 8 | 6 | 96 | .!!. .!!! !!!. .... | 2022 | | E2 | 4 |
| 195 | 9 | 12 | 192 | !!!. .!!! !!!. .... | 2023 | | O | 5 |
| 196 | 8 | 6 | 96 | !..! .!!! !!!. .... | 2025 | | E2 | 4 |
| 197 | 9 | 12 | 192 | !!.! .!!! !!!. .... | 2027 | | O | 5 |
| 198 | 10 | 6 | 96 | !!!! .!!! !!!. .... | 2031 | | E4 | 4 |
| 199 | 7 | 6 | 96 | .... !!!! !!!. .... | 2032 | | O | 1 |
| 200 | 8 | 12 | 192 | !... !!!! !!!. .... | 2033 | | E3 | 2 |
| 201 | 8 | 24 | 384 | .!.. !!!! !!!. .... | 2034 | | E2 | 2 |
| 202 | 9 | 24 | 384 | !!.. !!!! !!!. .... | 2035 | | O | 3 |
| 203 | 9 | 12 | 192 | .!!. !!!! !!!. .... | 2038 | | O | 3 |
| 204 | 10 | 12 | 192 | !!!. !!!! !!!. .... | 2039 | | E1 | 4 |
| 205 | 8 | 12 | 192 | ...! !!!! !!!. .... | 2040 | | E1 | 2 |
| 206 | 9 | 12 | 192 | !..! !!!! !!!. .... | 2041 | | O | 3 |
| 207 | 9 | 24 | 384 | .!.! !!!! !!!. .... | 2042 | | O | 3 |
| 208 | 10 | 24 | 384 | !!.! !!!! !!!. .... | 2043 | | E2 | 4 |
| 209 | 10 | 12 | 192 | .!!! !!!! !!!. .... | 2046 | | E3 | 4 |
| 210 | 11 | 12 | 192 | !!!! !!!! !!!. .... | 2047 | | O | 3 |
| 211 | 8 | 6 | 12 | .... !!!! !!!! .... | 4080 | N(3,2) | E0 | 0 |
| 212 | 9 | 6 | 96 | !... !!!! !!!! .... | 4081 | | O | 1 |
| 213 | 10 | 12 | 96 | !!.. !!!! !!!! .... | 4083 | | E1 | 2 |
| 214 | 10 | 6 | 48 | .!!. !!!! !!!! .... | 4086 | | E2 | 2 |
| 215 | 11 | 6 | 96 | !!!. !!!! !!!! .... | 4087 | | O | 3 |
| 216 | 12 | 6 | 24 | !!!! !!!! !!!! .... | 4095 | | E0 | 4 |
| 217 | 6 | 1 | 8 | ...! .!!. .!!. !... | 5736 | | E4 | 2 |
| 218 | 7 | 1 | 16 | !..! .!!. .!!. !... | 5737 | | O | 1 |
| 219 | 7 | 4 | 64 | .!.! .!!. .!!. !... | 5738 | | O | 3 |
| 220 | 8 | 4 | 64 | !!.! .!!. .!!. !... | 5739 | | E3 | 2 |
| 221 | 8 | 6 | 96 | .!!! .!!. .!!. !... | 5742 | | E2 | 4 |
| 222 | 9 | 6 | 96 | !!!! .!!. .!!. !... | 5743 | | O | 3 |
| 223 | 9 | 4 | 64 | .!!! !!!. .!!. !... | 5758 | | O | 5 |
| 224 | 10 | 4 | 64 | !!!! !!!. .!!. !... | 5759 | | E1 | 4 |
| 225 | 5 | 1 | 16 | !... ...! .!!. !... | 5761 | | O | 5 |
| 226 | 6 | 6 | 96 | !!.. ...! .!!. !... | 5763 | | E2 | 4 |
| 227 | 6 | 12 | 96 | .!!. ...! .!!. !... | 5766 | | E3 | 2 |
| 228 | 7 | 12 | 192 | !!!. ...! .!!. !... | 5767 | | O | 3 |
| 229 | 6 | 4 | 64 | !..! ...! .!!. !... | 5769 | | E3 | 4 |
| 230 | 7 | 12 | 192 | !!.! ...! .!!. !... | 5771 | | O | 5 |
| 231 | 7 | 12 | 192 | .!!! ...! .!!. !... | 5774 | | O | 3 |
| 232 | 8 | 12 | 192 | !!!! ...! .!!. !... | 5775 | | E1 | 4 |
| 233 | 7 | 4 | 64 | .!!. !..! .!!. !... | 5782 | | O | 1 |
| 234 | 8 | 4 | 64 | !!!. !..! .!!. !... | 5783 | | E4 | 2 |
| 235 | 6 | 6 | 96 | ...! !..! .!!. !... | 5784 | | E4 | 4 |
| 236 | 7 | 12 | 192 | !..! !..! .!!. !... | 5785 | | O | 3 |
| 237 | 7 | 24 | 384 | .!.! !..! .!!. !... | 5786 | | O | 3 |
| 238 | 8 | 24 | 384 | !!.! !..! .!!. !... | 5787 | | E3 | 4 |
| 239 | 8 | 12 | 192 | .!!! !..! .!!. !... | 5790 | | E2 | 2 |
| 240 | 9 | 12 | 192 | !!!! !..! .!!. !... | 5791 | | O | 3 |
| 241 | 7 | 6 | 96 | !..! .!.! .!!. !... | 5801 | | O | 3 |
| 242 | 8 | 6 | 96 | !!.! .!.! .!!. !... | 5803 | | E4 | 4 |
| 243 | 7 | 12 | 192 | ..!! .!.! .!!. !... | 5804 | | O | 5 |
| 244 | 8 | 24 | 384 | !.!! .!.! .!!. !... | 5805 | | E2 | 4 |
| 245 | 8 | 24 | 384 | .!!! .!.! .!!. !... | 5806 | | E3 | 4 |
| 246 | 9 | 24 | 384 | !!!! .!.! .!!. !... | 5807 | | O | 5 |
| 247 | 8 | 12 | 192 | ..!! !!.! .!!. !... | 5820 | | E1 | 4 |
| 248 | 9 | 12 | 192 | !.!! !!.! .!!. !... | 5821 | | O | 5 |
| 249 | 9 | 12 | 192 | .!!! !!.! .!!. !... | 5822 | | O | 3 |
| 250 | 10 | 12 | 192 | !!!! !!.! .!!. !... | 5823 | | E2 | 4 |
| 251 | 8 | 4 | 64 | !..! .!!! .!!. !... | 5865 | | E1 | 2 |
| 252 | 8 | 6 | 96 | .!.! .!!! .!!. !... | 5866 | | E2 | 4 |
| 253 | 9 | 12 | 192 | !!.! .!!! .!!. !... | 5867 | | O | 3 |
| 254 | 9 | 12 | 192 | .!!! .!!! .!!. !... | 5870 | | O | 5 |
| 255 | 10 | 12 | 192 | !!!! .!!! .!!. !... | 5871 | | E3 | 4 |
| 256 | 10 | 4 | 64 | .!!! !!!! .!!. !... | 5886 | | E4 | 4 |
| 257 | 11 | 4 | 64 | !!!! !!!! .!!. !... | 5887 | | O | 3 |
| 258 | 10 | 1 | 16 | .!!! !!!. !!!. !... | 6014 | | E0 | 6 |
| 259 | 11 | 1 | 16 | !!!! !!!. !!!. !... | 6015 | | O | 5 |
| 260 | 6 | 1 | 16 | !... ...! !!!. !... | 6017 | | E0 | 6 |
| 261 | 7 | 6 | 96 | !!.. ...! !!!. !... | 6019 | | O | 5 |
| 262 | 8 | 12 | 192 | !!!. ...! !!!. !... | 6023 | | E2 | 4 |
| 263 | 7 | 4 | 64 | !..! ...! !!!. !... | 6025 | | O | 5 |
| 264 | 8 | 12 | 192 | !!.! ...! !!!. !... | 6027 | | E1 | 4 |
| 265 | 8 | 12 | 96 | .!!! ...! !!!. !... | 6030 | | E0 | 4 |
| 266 | 9 | 12 | 192 | !!!! ...! !!!. !... | 6031 | | O | 3 |
| 267 | 8 | 4 | 64 | .!!. !..! !!!. !... | 6038 | | E3 | 2 |
| 268 | 9 | 4 | 64 | !!!. !..! !!!. !... | 6039 | | O | 3 |
| 269 | 7 | 12 | 192 | ...! !..! !!!. !... | 6040 | | O | 5 |
| 270 | 8 | 12 | 192 | !..! !..! !!!. !... | 6041 | | E3 | 4 |
| 271 | 8 | 24 | 384 | .!.! !..! !!!. !... | 6042 | | E2 | 4 |
| 272 | 9 | 24 | 384 | !!.! !..! !!!. !... | 6043 | | O | 5 |
| 273 | 9 | 12 | 192 | .!!! !..! !!!. !... | 6046 | | O | 3 |
| 274 | 10 | 12 | 192 | !!!! !..! !!!. !... | 6047 | | E1 | 4 |
| 275 | 8 | 6 | 96 | !..! .!.! !!!. !... | 6057 | | E2 | 4 |
| 276 | 9 | 6 | 96 | !!.! .!.! !!!. !... | 6059 | | O | 3 |
| 277 | 8 | 24 | 384 | ..!! .!.! !!!. !... | 6060 | | E1 | 4 |
| 278 | 9 | 24 | 384 | !.!! .!.! !!!. !... | 6061 | | O | 5 |
| 279 | 9 | 24 | 384 | .!!! .!.! !!!. !... | 6062 | | O | 3 |
| 280 | 10 | 24 | 384 | !!!! .!.! !!!. !... | 6063 | | E2 | 4 |
| 281 | 9 | 12 | 192 | ..!! !!.! !!!. !... | 6076 | | O | 5 |
| 282 | 10 | 12 | 192 | !.!! !!.! !!!. !... | 6077 | | E0 | 6 |
| 283 | 10 | 12 | 192 | .!!! !!.! !!!. !... | 6078 | | E1 | 4 |
| 284 | 11 | 12 | 192 | !!!! !!.! !!!. !... | 6079 | | O | 5 |
| 285 | 8 | 4 | 32 | ...! .!!! !!!. !... | 6120 | | E0 | 4 |
| 286 | 9 | 4 | 64 | !..! .!!! !!!. !... | 6121 | | O | 3 |
| 287 | 9 | 12 | 192 | .!.! .!!! !!!. !... | 6122 | | O | 3 |
| 288 | 10 | 12 | 192 | !!.! .!!! !!!. !... | 6123 | | E1 | 4 |
| 289 | 10 | 12 | 192 | .!!! .!!! !!!. !... | 6126 | | E2 | 4 |
| 290 | 11 | 12 | 192 | !!!! .!!! !!!. !... | 6127 | | O | 5 |
| 291 | 11 | 4 | 64 | .!!! !!!! !!!. !... | 6142 | | O | 5 |
| 292 | 12 | 4 | 64 | !!!! !!!! !!!. !... | 6143 | | E3 | 4 |
| 293 | 8 | 4 | 32 | !!!. .!!! ...! !... | 6375 | | E0 | 4 |
| 294 | 9 | 4 | 64 | !!!! .!!! ...! !... | 6383 | | O | 3 |
| 295 | 10 | 4 | 32 | !!!! !!!! ...! !... | 6399 | | E3 | 2 |
| 296 | 7 | 3 | 48 | !... .!!! !..! !... | 6625 | | O | 5 |
| 297 | 8 | 12 | 192 | !!.. .!!! !..! !... | 6627 | | E2 | 4 |
| 298 | 8 | 12 | 192 | .!!. .!!! !..! !... | 6630 | | E1 | 2 |
| 299 | 9 | 12 | 192 | !!!. .!!! !..! !... | 6631 | | O | 3 |
| 300 | 8 | 12 | 192 | !..! .!!! !..! !... | 6633 | | E1 | 4 |
| 301 | 8 | 12 | 192 | .!.! .!!! !..! !... | 6634 | | E2 | 4 |
| 302 | 9 | 24 | 384 | !!.! .!!! !..! !... | 6635 | | O | 5 |
| 303 | 9 | 12 | 192 | .!!! .!!! !..! !... | 6638 | | O | 3 |
| 304 | 10 | 12 | 192 | !!!! .!!! !..! !... | 6639 | | E3 | 4 |
| 305 | 8 | 3 | 48 | !... !!!! !..! !... | 6641 | | E4 | 4 |
| 306 | 9 | 12 | 192 | !!.. !!!! !..! !... | 6643 | | O | 5 |
| 307 | 9 | 12 | 192 | .!!. !!!! !..! !... | 6646 | | O | 3 |
| 308 | 10 | 12 | 192 | !!!. !!!! !..! !... | 6647 | | E2 | 4 |
| 309 | 8 | 6 | 96 | ...! !!!! !..! !... | 6648 | | E2 | 4 |
| 310 | 9 | 12 | 192 | !..! !!!! !..! !... | 6649 | | O | 3 |
| 311 | 9 | 24 | 384 | .!.! !!!! !..! !... | 6650 | | O | 5 |
| 312 | 10 | 24 | 384 | !!.! !!!! !..! !... | 6651 | | E3 | 4 |
| 313 | 10 | 12 | 192 | .!!! !!!! !..! !... | 6654 | | E4 | 4 |
| 314 | 11 | 12 | 192 | !!!! !!!! !..! !... | 6655 | | O | 3 |
| 315 | 9 | 12 | 192 | .!!. !.!! !!.! !... | 7126 | | O | 5 |
| 316 | 10 | 12 | 192 | !!!. !.!! !!.! !... | 7127 | | E0 | 6 |
| 317 | 8 | 3 | 24 | ...! !.!! !!.! !... | 7128 | | E0 | 4 |
| 318 | 9 | 12 | 192 | !..! !.!! !!.! !... | 7129 | | O | 3 |
| 319 | 10 | 12 | 192 | !!.! !.!! !!.! !... | 7131 | | E1 | 4 |
| 320 | 10 | 12 | 192 | .!!! !.!! !!.! !... | 7134 | | E2 | 4 |
| 321 | 11 | 12 | 192 | !!!! !.!! !!.! !... | 7135 | | O | 5 |
| 322 | 8 | 12 | 96 | ..!. .!!! !!.! !... | 7140 | | E0 | 4 |
| 323 | 9 | 24 | 384 | !.!. .!!! !!.! !... | 7141 | | O | 3 |
| 324 | 10 | 12 | 192 | !!!. .!!! !!.! !... | 7143 | | E1 | 4 |
| 325 | 9 | 24 | 384 | ..!! .!!! !!.! !... | 7148 | | O | 3 |
| 326 | 10 | 24 | 384 | !.!! .!!! !!.! !... | 7149 | | E2 | 4 |
| 327 | 10 | 24 | 384 | .!!! .!!! !!.! !... | 7150 | | E1 | 4 |
| 328 | 11 | 24 | 384 | !!!! .!!! !!.! !... | 7151 | | O | 5 |
| 329 | 10 | 12 | 192 | ..!! !!!! !!.! !... | 7164 | | E3 | 4 |
| 330 | 11 | 24 | 384 | !.!! !!!! !!.! !... | 7165 | | O | 5 |
| 331 | 12 | 12 | 192 | !!!! !!!! !!.! !... | 7167 | | E2 | 4 |
| 332 | 8 | 6 | 48 | !... .!!! .!!! !... | 7905 | | E0 | 4 |
| 333 | 9 | 12 | 192 | !!.. .!!! .!!! !... | 7907 | | O | 3 |
| 334 | 9 | 3 | 48 | .!!. .!!! .!!! !... | 7910 | | O | 5 |
| 335 | 10 | 6 | 96 | !!!. .!!! .!!! !... | 7911 | | E2 | 4 |
| 336 | 9 | 6 | 96 | !..! .!!! .!!! !... | 7913 | | O | 3 |
| 337 | 10 | 12 | 192 | !!.! .!!! .!!! !... | 7915 | | E1 | 4 |
| 338 | 10 | 3 | 48 | .!!! .!!! .!!! !... | 7918 | | E0 | 6 |
| 339 | 11 | 6 | 96 | !!!! .!!! .!!! !... | 7919 | | O | 5 |
| 340 | 9 | 6 | 96 | !... !!!! .!!! !... | 7921 | | O | 3 |
| 341 | 10 | 24 | 384 | !!.. !!!! .!!! !... | 7923 | | E2 | 4 |
| 342 | 10 | 12 | 192 | .!!. !!!! .!!! !... | 7926 | | E3 | 4 |
| 343 | 11 | 12 | 192 | !!!. !!!! .!!! !... | 7927 | | O | 5 |
| 344 | 10 | 12 | 192 | !..! !!!! .!!! !... | 7929 | | E3 | 4 |
| 345 | 10 | 12 | 192 | .!.! !!!! .!!! !... | 7930 | | E2 | 4 |
| 346 | 11 | 24 | 384 | !!.! !!!! .!!! !... | 7931 | | O | 5 |
| 347 | 11 | 12 | 192 | .!!! !!!! .!!! !... | 7934 | | O | 5 |
| 348 | 12 | 12 | 192 | !!!! !!!! .!!! !... | 7935 | | E1 | 4 |
| 349 | 10 | 6 | 48 | !... !!!! !!!! !... | 8177 | | E2 | 2 |
| 350 | 10 | 12 | 96 | .!.. !!!! !!!! !... | 8178 | | E3 | 2 |
| 351 | 11 | 12 | 192 | !!.. !!!! !!!! !... | 8179 | | O | 3 |
| 352 | 11 | 6 | 96 | .!!. !!!! !!!! !... | 8182 | | O | 3 |
| 353 | 12 | 6 | 96 | !!!. !!!! !!!! !... | 8183 | | E4 | 4 |
| 354 | 10 | 6 | 48 | ...! !!!! !!!! !... | 8184 | | E4 | 2 |
| 355 | 11 | 6 | 96 | !..! !!!! !!!! !... | 8185 | | O | 3 |
| 356 | 11 | 12 | 192 | .!.! !!!! !!!! !... | 8186 | | O | 3 |
| 357 | 12 | 12 | 192 | !!.! !!!! !!!! !... | 8187 | | E3 | 4 |
| 358 | 12 | 6 | 96 | .!!! !!!! !!!! !... | 8190 | | E2 | 4 |
| 359 | 13 | 6 | 96 | !!!! !!!! !!!! !... | 8191 | | O | 3 |
| 360 | 8 | 4 | 8 | !!.. ..!! ..!! !!.. | 15555 | N(3,3) | E0 | 0 |
| 361 | 9 | 4 | 64 | !!!. ..!! ..!! !!.. | 15559 | | O | 1 |
| 362 | 10 | 4 | 32 | !!!! ..!! ..!! !!.. | 15567 | | E1 | 2 |
| 363 | 10 | 12 | 96 | !!!. !.!! ..!! !!.. | 15575 | | E2 | 2 |
| 364 | 10 | 12 | 96 | !!.! !.!! ..!! !!.. | 15579 | | E3 | 2 |
| 365 | 11 | 12 | 192 | !!!! !.!! ..!! !!.. | 15583 | | O | 3 |
| 366 | 12 | 12 | 48 | !!!! !!!! ..!! !!.. | 15615 | | E0 | 4 |
| 367 | 10 | 4 | 64 | .!!. !.!! !.!! !!.. | 15830 | | E3 | 4 |
| 368 | 11 | 4 | 64 | !!!. !.!! !.!! !!.. | 15831 | | O | 3 |
| 369 | 10 | 6 | 96 | .!.! !.!! !.!! !!.. | 15834 | | E4 | 4 |
| 370 | 11 | 12 | 192 | !!.! !.!! !.!! !!.. | 15835 | | O | 3 |
| 371 | 11 | 12 | 192 | .!!! !.!! !.!! !!.. | 15838 | | O | 5 |
| 372 | 12 | 12 | 192 | !!!! !.!! !.!! !!.. | 15839 | | E3 | 4 |
| 373 | 11 | 12 | 192 | !.!! .!!! !.!! !!.. | 15853 | | O | 5 |
| 374 | 12 | 12 | 192 | !!!! .!!! !.!! !!.. | 15855 | | E2 | 4 |
| 375 | 12 | 6 | 96 | !.!! !!!! !.!! !!.. | 15869 | | E2 | 4 |
| 376 | 12 | 12 | 192 | .!!! !!!! !.!! !!.. | 15870 | | E1 | 4 |
| 377 | 13 | 12 | 192 | !!!! !!!! !.!! !!.. | 15871 | | O | 3 |
| 378 | 12 | 4 | 16 | ..!! !!!! !!!! !!.. | 16380 | | E0 | 4 |
| 379 | 13 | 4 | 64 | !.!! !!!! !!!! !!.. | 16381 | | O | 3 |
| 380 | 14 | 4 | 32 | !!!! !!!! !!!! !!.. | 16383 | | E1 | 2 |
| 381 | 8 | 1 | 2 | .!!. !..! !..! .!!. | 27030 | N(3,4) | E0 | 0 |
| 382 | 9 | 1 | 16 | !!!. !..! !..! .!!. | 27031 | | O | 1 |
| 383 | 10 | 6 | 48 | !!!! !..! !..! .!!. | 27039 | | E2 | 2 |
| 384 | 11 | 4 | 64 | !!!! !!.! !..! .!!. | 27071 | | O | 3 |
| 385 | 12 | 4 | 16 | !!!! !!!! !..! .!!. | 27135 | | E0 | 4 |
| 386 | 11 | 1 | 16 | !.!! !!.! !!.! .!!. | 27581 | | O | 5 |
| 387 | 12 | 1 | 16 | !!!! !!.! !!.! .!!. | 27583 | | E4 | 4 |
| 388 | 10 | 1 | 8 | .!!. !.!! !!.! .!!. | 27606 | | E4 | 2 |
| 389 | 11 | 3 | 48 | !!!. !.!! !!.! .!!. | 27607 | | O | 3 |
| 390 | 12 | 6 | 96 | !!!! !.!! !!.! .!!. | 27615 | | E2 | 4 |
| 391 | 12 | 4 | 64 | !.!! !!!! !!.! .!!. | 27645 | | E1 | 4 |
| 392 | 13 | 4 | 64 | !!!! !!!! !!.! .!!. | 27647 | | O | 3 |
| 393 | 12 | 3 | 12 | .!!. !!!! !!!! .!!. | 28662 | | E0 | 4 |
| 394 | 13 | 3 | 48 | !!!. !!!! !!!! .!!. | 28663 | | O | 3 |
| 395 | 12 | 6 | 24 | !..! !!!! !!!! .!!. | 28665 | | E0 | 4 |
| 396 | 13 | 6 | 96 | !!.! !!!! !!!! .!!. | 28667 | | O | 3 |
| 397 | 14 | 6 | 48 | !!!! !!!! !!!! .!!. | 28671 | | E2 | 2 |
| 398 | 14 | 4 | 32 | !!!! !!!! .!!! !!!. | 32511 | | E3 | 2 |
| 399 | 14 | 1 | 8 | .!!! !!!! !!!! !!!. | 32766 | | E4 | 2 |
| 400 | 15 | 1 | 16 | !!!! !!!! !!!! !!!. | 32767 | | O | 1 |
| 401 | 16 | 1 | 1 | !!!! !!!! !!!! !!!! | 65535 | N(4,1) | E0 | 0 |

## Greatest big equivalence classes (ggbec)

Table of the 39 ggbecs :
| nonlin | wec | | | | |
| 0 | E0 | 32 | = | 2$^5$ | &0000ggbec 0 |
| 1 | O | 512 | = | 2$^9$ | &0001ggbec 1 |
| 2 | E1 | 1024 | = | 2$^{10}$ | &0003ggbec 3 |
| 2 | E2 | 1536 | = | 2$^9$ * 3 | &0006ggbec 6 |
| 3 | O | 3072 | = | 2$^{10}$ * 3 | &0007ggbec 7 |
| 4 | E0 | 192 | = | 2$^6$ * 3 | &0015ggbec 15 |
| 3 | O | 2048 | = | 2$^{11}$ | &0022ggbec 22 |
| 4 | E3 | 1024 | = | 2$^{10}$ | &0023ggbec 23 |
| 2 | E3 | 1024 | = | 2$^{10}$ | &0024ggbec 24 |
| 3 | O | 6144 | = | 2$^{11}$ * 3 | &0025ggbec 25 |
| 4 | E2 | 3072 | = | 2$^{10}$ * 3 | &0027ggbec 27 |
| 4 | E1 | 3072 | = | 2$^{10}$ * 3 | &0030ggbec 30 |
| 4 | E0 | 384 | = | 2$^7$ * 3 | &0060ggbec 60 |
| 4 | E0 | 128 | = | 2$^7$ | &0105ggbec 105 |
| 4 | E4 | 256 | = | 2$^8$ | &0278ggbec 278 |
| 5 | O | 512 | = | 2$^9$ | &0279ggbec 279 |
| 3 | O | 3072 | = | 2$^{10}$ * 3 | &0280ggbec 280 |
| 4 | E4 | 1536 | = | 2$^9$ * 3 | &0281ggbec 281 |
| 4 | E3 | 3072 | = | 2$^{10}$ * 3 | &0282ggbec 282 |
| 5 | O | 6144 | = | 2$^{11}$ * 3 | &0283ggbec 283 |
| 5 | O | 1536 | = | 2$^9$ * 3 | &0286ggbec 286 |
| 4 | E2 | 1536 | = | 2$^9$ * 3 | &0287ggbec 287 |
| 4 | E2 | 3072 | = | 2$^{10}$ * 3 | &0300ggbec 300 |
| 5 | O | 6144 | = | 2$^{11}$ * 3 | &0301ggbec 301 |
| 4 | E3 | 3072 | = | 2$^{10}$ * 3 | &0303ggbec 303 |
| 4 | E1 | 3072 | = | 2$^{10}$ * 3 | &0317ggbec 317 |
| 4 | E2 | 1536 | = | 2$^9$ * 3 | &0318ggbec 318 |
| 3 | O | 2048 | = | 2$^{11}$ | &0319ggbec 319 |
| 4 | E1 | 1024 | = | 2$^{10}$ | &0360ggbec 360 |
| 4 | E2 | 1536 | = | 2$^9$ * 3 | &0363ggbec 363 |
| 3 | O | 1536 | = | 2$^9$ * 3 | &0367ggbec 367 |
| 2 | E4 | 256 | = | 2$^8$ | &0383ggbec 383 |
| 4 | E0 | 128 | = | 2$^7$ | &0831ggbec 831 |
| 6 | E0 | 96 | = | 2$^5$ * 3 | &0854ggbec 854 |
| 6 | E0 | 384 | = | 2$^7$ * 3 | &0857ggbec 857 |
| 6 | E0 | 384 | = | 2$^7$ * 3 | &0874ggbec 874 |
| 4 | E0 | 96 | = | 2$^5$ * 3 | &1632ggbec 1632 |
| 4 | E0 | 192 | = | 2$^6$ * 3 | &1647ggbec 1647 |
| 6 | E0 | 32 | = | 2$^5$ | &6014ggbec 6014 |

## Walsh equivalence classes (wec)

| 32768 = 2$^{15}$ | wec O |
| 2048 = 2$^{11}$ | wec E0 |
| 8192 = 2$^{13}$ | wec E1 |
| 12288 = 2$^{12}$ * 3 | wec E2 |
| 8192 = 2$^{13}$ | wec E3 |
| 2048 = 2$^{11}$ | wec E4 |

## Ring count vectors (rcv)

A step back: The 16 2-ary Boolean functions correspond to the subsets of the nested set or hereditarily finite set V$_3$ = P$^3$() = {  , {} , {{}} , {,{}} }. That is hard to read, so the nested sets are represented by nested rings of different size.
In the same way every 4-ary Boolean function corresponds to a subset of V$_4$ = P$^4$():
This set corresponds to the tautology. The cardinality of the subsets of V$_4$ corresponds to the weight of the functions, in this case 16. Without the outer ring there are four different sizes of rings in these graphics. By counting also the smaller rings, the concept of weight can be refined, which leads to a 4-place ring count vector (rcv) for every 4-ary Boolean function. For the tautology it is (16,32,32,16).
For the next function the rcv is (8,16,16,8):
The vectors are not always so regular. For the next function it is (7,10,8,4):
Every set of 4-ary Boolean functions has a 5-place rcv where the first digit is the set's cardinality. E.g. the set that contains only the above-mentioned function has the rcv (1,7,10,8,4).
Sets of functions that are of some interest are equivalence classes. The last four digits in the rcvs of equivalence classes usually have the pattern n,2n,2n,n.
In the ggbec table above only the first digits are shown, because the rcvs have the pattern n,8n,16n,16n,8n. This is also true for the wec table, because a wec is a union of many ggbecs.
Every rcv has a leading one for the surrounding set, but for the sake of brevity it was omitted here. With it the sums are  A116549.

## Subgroups of nimber addition

       See: Subgroups of nimber addition#Z$_2$$^4$

## Bent functions

There are 896 4-ary Bent functions, which belong to four different greater big equivalence classes (gbecs).
Their nonlinearity is $2^{4-1}-2^{{\frac {4}{2}}-1}=8-2=6$ and their Hamming weight is $2^{4-1}\pm 2^{{\frac {4}{2}}-1}=8\pm 2$ .
Only the functions with Hamming weight 6 are shown. Those with Hamming weight 10 are their complements.
| Belongs to BEC 391 with 16 functions. | The simplest example of a 4-ary bent function: $x_{1}\land x_{2}~~\oplus ~~x_{3}\land x_{4}$ Belongs to BEC 381 with 3*16 functions. | Belongs to BEC 382 with 12*16 functions. | Belongs to BEC 383 with 12*16 functions. |
Big equivalence classes (becs) :
| BEC 391 with 16 functions | BEC 381 with 3*16 = 48 functions | BEC 382 with 12*16 = 192 functions | BEC 383 with 12*16 = 192 functions |
It's worth taking a look at the distribution of dots in the small matrices in the bec files. They are all closely related to permuted Walsh matrices, which is not the case for any other 4-ary Boolean functions. Even more: Only the bent functions have 8 dots in every row (except the top row) of the small matrices. Walsh permutation; bent functions

## About the bec files

The files in this category describe big equivalence classes (becs). They contain 24 sections, showing 16×16 sec matrices and a number between 0 and 23, representing a permutation of the four arguments. The big matrices and the small matrices on the right show the same background colors representing binary numbers, but they differ in the dots. In the big matrices the dots show which fields differ from the top row in the top matrix, while in the small matrices the dots show which fields differ from the top row of the matrix itself. The numbers in the small left matrices show how the bits of the Boolean function in the top row of the top matrix are permuted. Their top rows are bit permutations.