# 4-ary Boolean functions; rational bec table

Compare 4-ary Boolean functions#Big equivalence classes (bec)

In this table the Boolean functions are sorted as fractions in colex order.
This order is independent of arity (unlike that of integers in ).
The number of BECs by actual arity is found in . (Compare for acual or smaller arity.)

```        n        0,  1,  2,   3,    4        arity
2^n        1,  2,  4,   8,   16        period length
A000618(n)       2,  1,  3,  16,  380        number of BECs of arity = n
A000616(n)       2,  3,  6,  22,  402        number of BECs of arity ≤ n
```

The same information in a Python dict: https://pastebin.com/q2d4aWVE
Map from the 65536 functions to the BEC index numbers: https://pastebin.com/3fBShT7M
The 168 monotonic functions belonging to 30 BECs: https://pastebin.com/JUeKsasG

I accidentally reflected the period in the table below. E.g. 1/3 in binary is .0101... - not .1010. I will fix it later.

c a weight so
na
no
mo
rational number value f/s

s/b
ary
f/b
ary
s/b f/b
num over den period num den dec 0 1 2 3 4 0 1 2 3 4

0 1 0 0 / 1 0 0 0000000000000000 0 1 0.000000 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 0 1 / 1 16 s 0 1111111111111111 1 1 1.000000 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 / 2 8 s 0 1010101010101010 1 3 0.333333 2 1 2 1 2 3 4 2 4 6 8
3 2 1 / 2 8 s 4 0110011001100110 2 5 0.400000 2 1 2 1 3 6 2 6 12
4 5 2 1 / 4 4 s 0 1000100010001000 1 15 0.066667 4 1 4 1 3 6 4 12 24
5 4 2 3 / 4 12 0 1110111011101110 7 15 0.466667 4 1 4 1 3 6 4 12 24
6 3 1 / 2 8 2 0111100001111000 2 17 0.117647 8 3 24 3 12 24 96
7 3 1 / 2 8 s 4 1001011010010110 7 17 0.411765 2 1 2 1 4 2 8
8 13 3 1 / 4 4 s 2 0010100000101000 4 51 0.078431 4 3 12 3 12 12 48
9 12 3 3 / 8 6 2 1001100010011000 5 51 0.098039 8 3 24 3 12 24 96
10 11 3 3 / 8 6 0 1100010011000100 7 51 0.137255 8 3 24 3 12 24 96
11 10 3 5 / 8 10 0 1110110011101100 11 51 0.215686 8 3 24 3 12 24 96
12 9 3 5 / 8 10 2 0111011001110110 22 51 0.431373 8 3 24 3 12 24 96
13 8 3 3 / 4 12 2 1011111010111110 25 51 0.490196 4 3 12 3 12 12 48
14 16 3 1 / 4 4 s 2 0001100000011000 8 85 0.094118 4 1 4 1 4 4 16
15 3 1 / 2 8 2 1101100011011000 9 85 0.105882 8 3 24 3 12 24 96
16 14 3 3 / 4 12 2 0111111001111110 42 85 0.494118 4 1 4 1 4 4 16
17 21 3 1 / 8 2 s 0 1000000010000000 1 255 0.003922 8 1 8 1 4 8 32
18 20 3 3 / 8 6 2 0110100001101000 22 255 0.086275 8 1 8 1 4 8 32
19 3 1 / 2 8 0 1110100011101000 23 255 0.090196 8 1 8 1 4 8 32
20 18 3 5 / 8 10 2 1101011011010110 107 255 0.419608 8 1 8 1 4 8 32
21 17 3 7 / 8 14 0 1111111011111110 127 255 0.498039 8 1 8 1 4 8 32
22 4 1 / 2 8 1 0111111110000000 2 257 0.007782 16 4 64 4 64
23 4 1 / 2 8 3 1001111101100000 7 257 0.027237 8 12 96 12 96
24 4 1 / 2 8 3 0001111111100000 8 257 0.031128 16 12 192 12 192
25 4 1 / 2 8 4 1001011101101000 23 257 0.089494 16 4 64 4 64
26 4 1 / 2 8 4 0001011111101000 24 257 0.093385 8 4 32 4 32
27 4 1 / 2 8 2 1110011100011000 25 257 0.097276 8 4 32 4 32
28 4 1 / 2 8 3 0110011110011000 26 257 0.101167 16 12 192 12 192
29 4 1 / 2 8 4 0010011111011000 28 257 0.108949 8 12 96 12 96
30 4 1 / 2 8 4 1000011101111000 31 257 0.120623 8 6 48 6 48
31 4 1 / 2 8 s 5 0110100110010110 106 257 0.412451 2 1 2 1 2
32 95 4 1 / 4 4 1 0010101010000000 4 771 0.005188 16 12 192 12 192
33 94 4 5 / 16 5 2 1001010110000000 5 771 0.006485 16 12 192 12 192
34 109 4 5 / 16 5 2 1100101001000000 7 771 0.009079 16 12 192 12 192
35 96 4 7 / 16 7 1 1110010111000000 11 771 0.014267 16 24 384 24 384
36 44 4 7 / 16 7 1 1111001001100000 19 771 0.024643 16 12 192 12 192
37 97 4 5 / 16 5 3 0010010101100000 20 771 0.025940 16 12 192 12 192
38 67 4 7 / 16 7 2 0111001011100000 22 771 0.028534 16 12 192 12 192
39 98 4 7 / 16 7 2 1100010111100000 23 771 0.029831 16 12 192 12 192
40 78 4 5 / 16 5 1 1011001000010000 25 771 0.032425 16 12 192 12 192
41 82 4 5 / 16 5 2 0011001010010000 28 771 0.036316 16 24 384 24 384
42 91 4 3 / 8 6 1 1101001001010000 31 771 0.040208 16 24 384 24 384
43 93 4 3 / 8 6 1 0101001011010000 34 771 0.044099 16 12 192 12 192
44 36 4 9 / 16 9 1 1111100111010000 35 771 0.045396 16 12 192 12 192
45 90 4 5 / 16 5 1 1001001000110000 37 771 0.047990 16 6 96 6 96
46 57 4 7 / 16 7 1 0111100100110000 38 771 0.049287 16 12 192 12 192
47 65 4 1 / 2 8 1 1011100110110000 41 771 0.053178 16 12 192 12 192
48 72 4 7 / 16 7 2 0011100101110000 44 771 0.057069 16 24 384 24 384
49 50 4 7 / 16 7 2 0110001011110000 46 771 0.059663 16 12 192 12 192
50 49 4 9 / 16 9 2 1101100111110000 47 771 0.060960 16 12 192 12 192
51 110 4 5 / 16 5 2 1100001001001000 55 771 0.071336 16 24 384 24 384
52 111 4 5 / 16 5 2 0100001011001000 58 771 0.075227 16 24 384 24 384
53 99 4 1 / 2 8 1 1110100111001000 59 771 0.076524 16 12 192 12 192
54 107 4 1 / 4 4 s 2 1000001000101000 61 771 0.079118 4 4 16 4 16
55 100 4 3 / 8 6 3 0110100100101000 62 771 0.080415 8 12 96 12 96
56 101 4 7 / 16 7 1 1010100110101000 65 771 0.084306 16 12 192 12 192
57 46 4 9 / 16 9 1 1111110001101000 67 771 0.086900 16 12 192 12 192
58 102 4 3 / 8 6 3 0010100101101000 68 771 0.088197 8 12 96 12 96
59 69 4 9 / 16 9 1 0111110011101000 70 771 0.090791 16 6 96 6 96
60 103 4 1 / 2 8 1 1100100111101000 71 771 0.092088 16 12 192 12 192
61 79 4 7 / 16 7 3 1011110000011000 73 771 0.094682 16 24 384 24 384
62 104 4 5 / 16 5 3 0100100100011000 74 771 0.095979 16 12 192 12 192
63 87 4 7 / 16 7 3 0011110010011000 76 771 0.098573 16 24 384 24 384
64 105 4 3 / 8 6 2 1000100110011000 77 771 0.099870 8 6 48 6 48
65 47 4 1 / 2 8 1 1101110001011000 79 771 0.102464 16 12 192 12 192
66 71 4 1 / 2 8 2 0101110011011000 82 771 0.106355 16 12 192 12 192
67 38 4 9 / 16 9 2 1111000111011000 83 771 0.107652 16 12 192 12 192
68 89 4 7 / 16 7 3 1001110000111000 85 771 0.110246 16 12 192 12 192
69 59 4 7 / 16 7 1 0111000100111000 86 771 0.111543 16 6 96 6 96
70 81 4 7 / 16 7 4 0001110010111000 88 771 0.114137 16 12 192 12 192
71 66 4 1 / 2 8 2 1011000110111000 89 771 0.115435 16 12 192 12 192
72 48 4 9 / 16 9 2 1110110001111000 91 771 0.118029 16 24 384 24 384
73 86 4 3 / 8 6 1 0100110011000100 106 771 0.137484 8 12 96 12 96
74 106 4 7 / 16 7 2 1110000111000100 107 771 0.138781 16 12 192 12 192
75 80 4 3 / 8 6 2 1011010000010100 121 771 0.156939 16 24 384 24 384
76 88 4 3 / 8 6 3 0011010010010100 124 771 0.160830 16 24 384 24 384
77 92 4 7 / 16 7 0 1101010001010100 127 771 0.164721 16 12 192 12 192
78 40 4 11 / 16 11 1 1111111011010100 131 771 0.169909 16 12 192 12 192
79 61 4 9 / 16 9 3 0111111000110100 134 771 0.173800 16 24 384 24 384
80 75 4 5 / 8 10 2 1011111010110100 137 771 0.177691 16 24 384 24 384
81 70 4 9 / 16 9 4 0011111001110100 140 771 0.181582 16 12 192 12 192
82 41 4 11 / 16 11 2 1101111011110100 143 771 0.185473 16 24 384 24 384
83 4 1 / 2 8 2 0110111000101100 158 771 0.204929 16 24 384 24 384
84 108 4 9 / 16 9 1 1010111010101100 161 771 0.208820 16 24 384 24 384
85 4 1 / 2 8 3 0010111001101100 164 771 0.212711 16 24 384 24 384
86 73 4 5 / 8 10 1 1100111011101100 167 771 0.216602 8 12 96 12 96
87 63 4 9 / 16 9 3 0111011000111100 182 771 0.236057 16 24 384 24 384
88 76 4 5 / 8 10 3 1011011010111100 185 771 0.239948 16 24 384 24 384
89 68 4 9 / 16 9 3 0011011001111100 188 771 0.243839 16 12 192 12 192
90 45 4 11 / 16 11 1 1101011011111100 191 771 0.247730 16 6 96 6 96
91 42 4 5 / 8 10 1 1111101011010010 227 771 0.294423 16 24 384 24 384
92 77 4 9 / 16 9 0 1011101010110010 233 771 0.302205 16 12 192 12 192
93 43 4 5 / 8 10 1 1101101011110010 239 771 0.309987 16 12 192 12 192
94 33 4 11 / 16 11 2 0111111101101010 262 771 0.339818 16 12 192 12 192
95 32 4 3 / 4 12 1 1011111111101010 265 771 0.343709 16 12 192 12 192
96 35 4 9 / 16 9 1 0011111100011010 268 771 0.347601 16 24 384 24 384
97 37 4 11 / 16 11 3 1001111111011010 277 771 0.359274 16 12 192 12 192
98 39 4 9 / 16 9 2 0001111100111010 280 771 0.363165 16 12 192 12 192
99 53 4 1 / 2 8 1 0011011100010110 316 771 0.409857 16 12 192 12 192
100 55 4 5 / 8 10 3 1101011110010110 319 771 0.413748 8 12 96 12 96
101 56 4 9 / 16 9 1 0101011101010110 322 771 0.417639 16 12 192 12 192
102 58 4 5 / 8 10 3 1001011111010110 325 771 0.421530 8 12 96 12 96
103 60 4 1 / 2 8 1 0001011100110110 328 771 0.425422 16 12 192 12 192
104 62 4 11 / 16 11 3 1110011110110110 331 771 0.429313 16 12 192 12 192
105 64 4 5 / 8 10 2 0110011101110110 334 771 0.433204 8 6 48 6 48
106 74 4 9 / 16 9 2 0011101100011110 364 771 0.472114 16 12 192 12 192
107 54 4 3 / 4 12 2 1110101110111110 379 771 0.491569 4 4 16 4 16
108 84 4 7 / 16 7 1 0101001101010001 418 771 0.542153 16 24 384 24 384
109 34 4 11 / 16 11 2 1011111100110101 521 771 0.675746 16 12 192 12 192
110 51 4 11 / 16 11 2 1011011100111101 569 771 0.738003 16 24 384 24 384
111 52 4 11 / 16 11 2 0011011110111101 572 771 0.741894 16 24 384 24 384
112 180 4 1 / 4 4 2 0001100110000000 8 1285 0.006226 16 12 192 12 192
113 148 4 3 / 8 6 2 1101001110000000 9 1285 0.007004 16 24 384 24 384
114 181 4 3 / 8 6 2 1110100101000000 13 1285 0.010117 16 12 192 12 192
115 182 4 3 / 8 6 2 0110100111000000 18 1285 0.014008 16 12 192 12 192
116 171 4 3 / 8 6 3 1001001111000000 19 1285 0.014786 16 12 192 12 192
117 130 4 3 / 8 6 1 1111010000100000 21 1285 0.016342 16 6 96 6 96
118 166 4 3 / 8 6 2 1011010001100000 31 1285 0.024125 16 12 192 12 192
119 185 4 1 / 4 4 s 3 0000011001100000 32 1285 0.024903 4 3 12 3 12
120 184 4 3 / 8 6 3 1100100101100000 33 1285 0.025681 16 12 192 12 192
121 154 4 3 / 8 6 2 0011010011100000 36 1285 0.028016 16 12 192 12 192
122 176 4 3 / 8 6 3 0100100111100000 38 1285 0.029572 16 24 384 24 384
123 144 4 3 / 8 6 2 0111101000010000 42 1285 0.032685 16 24 384 24 384
124 173 4 3 / 8 6 3 0011101001010000 52 1285 0.040467 16 12 192 12 192
125 4 1 / 2 8 2 1101101011010000 57 1285 0.044358 16 24 384 24 384
126 177 4 3 / 8 6 3 1000001111010000 59 1285 0.045914 16 12 192 12 192
127 158 4 3 / 8 6 1 1011000100110000 63 1285 0.049027 16 12 192 12 192
128 168 4 3 / 8 6 2 0110010010110000 66 1285 0.051362 16 4 64 4 64
129 151 4 7 / 16 7 1 1001101010110000 67 1285 0.052140 16 12 192 12 192
130 117 4 5 / 8 10 1 1111110110110000 69 1285 0.053696 16 6 96 6 96
131 153 4 3 / 8 6 2 0001101001110000 72 1285 0.056031 16 12 192 12 192
132 152 4 7 / 16 7 1 1101000101110000 73 1285 0.056809 16 12 192 12 192
133 143 4 1 / 2 8 2 1001110111001000 99 1285 0.077043 16 24 384 24 384
134 186 4 5 / 16 5 2 0100101000101000 102 1285 0.079377 16 12 192 12 192
135 174 4 3 / 8 6 3 0001110100101000 104 1285 0.080934 16 12 192 12 192
136 167 4 7 / 16 7 2 1011100001101000 111 1285 0.086381 16 24 384 24 384
137 187 4 3 / 8 6 2 1100000101101000 113 1285 0.087938 16 6 96 6 96
138 4 1 / 2 8 3 0110110101101000 114 1285 0.088716 16 12 192 12 192
139 141 4 7 / 16 7 2 0011100011101000 116 1285 0.090272 16 24 384 24 384
140 145 4 3 / 8 6 2 0100000111101000 118 1285 0.091829 16 12 192 12 192
141 139 4 9 / 16 9 2 1010110111101000 119 1285 0.092607 16 24 384 24 384
142 188 4 1 / 4 4 s 3 1000000100011000 123 1285 0.095720 4 6 24 6 24
143 133 4 1 / 2 8 2 1100110110011000 129 1285 0.100389 16 24 384 24 384
144 123 4 5 / 8 10 2 1111111001011000 133 1285 0.103502 16 24 384 24 384
145 140 4 5 / 8 10 2 0111111011011000 138 1285 0.107393 16 12 192 12 192
146 4 1 / 2 8 3 1000110111011000 139 1285 0.108171 8 12 96 12 96
147 160 4 9 / 16 9 2 1011111000111000 143 1285 0.111284 16 6 96 6 96
148 113 4 5 / 8 10 2 1111010110111000 149 1285 0.115953 16 24 384 24 384
149 150 4 3 / 8 6 2 0001001001111000 152 1285 0.118288 16 12 192 12 192
150 149 4 5 / 8 10 2 1101111001111000 153 1285 0.119066 16 12 192 12 192
151 129 4 9 / 16 9 1 0111010101111000 154 1285 0.119844 16 12 192 12 192
152 132 4 9 / 16 9 1 1110001011111000 157 1285 0.122179 16 12 192 12 192
153 131 4 5 / 8 10 2 0101111011111000 158 1285 0.122957 16 12 192 12 192
154 121 4 5 / 8 10 2 1011010111111000 159 1285 0.123735 16 12 192 12 192
155 189 4 7 / 16 7 3 0010111010100100 188 1285 0.146304 16 24 384 24 384
156 4 1 / 2 8 3 1110010110100100 189 1285 0.147082 16 12 192 12 192
157 178 4 7 / 16 7 2 0110010101100100 194 1285 0.150973 16 24 384 24 384
158 127 4 5 / 8 10 1 1111110011100100 197 1285 0.153307 16 12 192 12 192
159 4 1 / 2 8 3 1010010111100100 199 1285 0.154864 16 24 384 24 384
160 147 4 7 / 16 7 2 0111110000010100 202 1285 0.157198 16 6 96 6 96
161 163 4 1 / 2 8 2 1011110010010100 207 1285 0.161089 16 12 192 12 192
162 175 4 7 / 16 7 2 0011110001010100 212 1285 0.164981 16 12 192 12 192
163 161 4 1 / 2 8 2 1011011000110100 223 1285 0.173541 16 12 192 12 192
164 165 4 1 / 2 8 3 1001110010110100 227 1285 0.176654 16 12 192 12 192
165 164 4 1 / 2 8 3 0011011010110100 228 1285 0.177432 16 12 192 12 192
166 118 4 5 / 8 10 2 1111100110110100 229 1285 0.178210 16 12 192 12 192
167 136 4 9 / 16 9 2 1101011001110100 233 1285 0.181323 16 24 384 24 384
168 128 4 5 / 8 10 2 1011100111110100 239 1285 0.185992 16 4 64 4 64
169 190 4 7 / 16 7 2 1010011000101100 263 1285 0.204669 16 12 192 12 192
170 179 4 3 / 8 6 3 0001100100101100 264 1285 0.205447 16 12 192 12 192
171 116 4 5 / 8 10 3 0011111101101100 276 1285 0.214786 16 12 192 12 192
172 183 4 3 / 8 6 3 0010100100011100 284 1285 0.221012 16 12 192 12 192
173 124 4 5 / 8 10 3 1111101001011100 293 1285 0.228016 16 12 192 12 192
174 135 4 5 / 8 10 3 0111101011011100 298 1285 0.231907 16 12 192 12 192
175 162 4 9 / 16 9 2 1011101000111100 303 1285 0.235798 16 12 192 12 192
176 122 4 5 / 8 10 3 1101101001111100 313 1285 0.243580 16 24 384 24 384
177 126 4 5 / 8 10 3 0010111101111100 316 1285 0.245914 16 12 192 12 192
178 157 4 9 / 16 9 2 0110111001101010 434 1285 0.337743 16 24 384 24 384
179 170 4 5 / 8 10 3 1101001111100110 521 1285 0.405447 16 12 192 12 192
180 112 4 3 / 4 12 2 0111111111100110 522 1285 0.406226 16 12 192 12 192
181 114 4 5 / 8 10 2 1011111100010110 527 1285 0.410117 16 12 192 12 192
182 115 4 5 / 8 10 2 0011111110010110 532 1285 0.414008 16 12 192 12 192
183 172 4 5 / 8 10 3 1110001111010110 541 1285 0.421012 16 12 192 12 192
184 120 4 5 / 8 10 3 1001111100110110 547 1285 0.425681 16 12 192 12 192
185 119 4 3 / 4 12 3 0110111111110110 562 1285 0.437354 4 3 12 3 12
186 134 4 11 / 16 11 2 1101011101011110 617 1285 0.480156 16 12 192 12 192
187 137 4 5 / 8 10 2 1001011100111110 627 1285 0.487938 16 6 96 6 96
188 142 4 3 / 4 12 3 1110011101111110 637 1285 0.495720 4 6 24 6 24
189 155 4 9 / 16 9 3 0101101111010001 702 1285 0.546304 16 24 384 24 384
190 169 4 9 / 16 9 2 1101001101011001 777 1285 0.604669 16 12 192 12 192
191 290 4 1 / 8 2 s 1 0000100010000000 16 3855 0.004150 8 6 48 6 48
192 285 4 3 / 16 3 1 1000010010000000