Linear Boolean functions
| Studies of Boolean functions |
Linear Boolean functions are Walsh functions and their negations.
A Walsh function is a variadic XOR (a.k.a. parity function).
The set of arguments in that XOR can easily be expressed as an integer, which shall be called Walsh index.
The natural way to denote a linear function is as Walsh index and parity. Walsh functions have parity 0. Their negations have parity 1.
Sometimes a more arcane notation is useful, namely as leader and quadrant.
The leader is the right shift of the Walsh index. The quadrant is the first and last digit of the truth table expressed in an integer.
The tables below show the Walsh indices on gray and the quadrants on colored background.
The Walsh indices of functions with quadrants 0 and 3 are evil numbers 0, 3, 5, 6... For quadrants 1 and 2 they are odious numbers 1, 2, 4, 7...
The columns marked with Ж show the Zhegalkin indices. Those of the Walsh functions are sequence 
A358126. Those of the negations are bigger by 1.
See also Linear and noble Boolean functions.
2-ary
[edit | edit source]| matrix | |
|---|---|
 |
 |
| Walsh weight |
Walsh index |
leader | Walsh | ¬ Walsh | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Q | Ж | Q | Ж | |||||||
| 0 | 0 | 0 | 0 | 
|
0 | 0 | 3 | 
|
15 | 1 |
| 1 | 1 | 0 | 2 | 
|
10 | 2 | 1 | 
|
5 | 3 |
| 1 | 2 | 1 | 2 | 
|
12 | 4 | 1 | 
|
3 | 5 |
| 2 | 3 | 1 | 0 | 
|
6 | 6 | 3 | 
|
9 | 7 |
3-ary
[edit | edit source]| matrix | |
|---|---|
 |
 |
| Walsh weight |
Walsh index |
leader | Walsh | ¬ Walsh | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Q | Ж | Q | Ж | |||||||
| 0 | 0 | 0 | 0 | 
|
0 | 0 | 3 | 
|
255 | 1 |
| 1 | 1 | 0 | 2 | 
|
170 | 2 | 1 | 
|
85 | 3 |
| 1 | 2 | 1 | 2 | 
|
204 | 4 | 1 | 
|
51 | 5 |
| 2 | 3 | 1 | 0 | 
|
102 | 6 | 3 | 
|
153 | 7 |
| 1 | 4 | 2 | 2 | 
|
240 | 16 | 1 | 
|
15 | 17 |
| 2 | 5 | 2 | 0 | 
|
90 | 18 | 3 | 
|
165 | 19 |
| 2 | 6 | 3 | 0 | 
|
60 | 20 | 3 | 
|
195 | 21 |
| 3 | 7 | 3 | 2 | 
|
150 | 22 | 1 | 
|
105 | 23 |
4-ary
[edit | edit source]
| matrix | ||||
|---|---|---|---|---|
|
| Walsh weight |
Walsh index |
leader | Walsh | ¬ Walsh | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Q | Ж | Q | Ж | |||||||
| 0 | 0 | 0 | 0 | 
|
0 | 0 | 3 | 
|
65535 | 1 |
| 1 | 1 | 0 | 2 | 
|
43690 | 2 | 1 | 
|
21845 | 3 |
| 1 | 2 | 1 | 2 | 
|
52428 | 4 | 1 | 
|
13107 | 5 |
| 2 | 3 | 1 | 0 | 
|
26214 | 6 | 3 | 
|
39321 | 7 |
| 1 | 4 | 2 | 2 | 
|
61680 | 16 | 1 | 
|
3855 | 17 |
| 2 | 5 | 2 | 0 | 
|
23130 | 18 | 3 | 
|
42405 | 19 |
| 2 | 6 | 3 | 0 | 
|
15420 | 20 | 3 | 
|
50115 | 21 |
| 3 | 7 | 3 | 2 | 
|
38550 | 22 | 1 | 
|
26985 | 23 |
| 1 | 8 | 4 | 2 | 
|
65280 | 256 | 1 | 
|
255 | 257 |
| 2 | 9 | 4 | 0 | 
|
21930 | 258 | 3 | 
|
43605 | 259 |
| 2 | 10 | 5 | 0 | 
|
13260 | 260 | 3 | 
|
52275 | 261 |
| 3 | 11 | 5 | 2 | 
|
39270 | 262 | 1 | 
|
26265 | 263 |
| 2 | 12 | 6 | 0 | 
|
4080 | 272 | 3 | 
|
61455 | 273 |
| 3 | 13 | 6 | 2 | 
|
42330 | 274 | 1 | 
|
23205 | 275 |
| 3 | 14 | 7 | 2 | 
|
49980 | 276 | 1 | 
|
15555 | 277 |
| 4 | 15 | 7 | 0 | 
|
27030 | 278 | 3 | 
|
38505 | 279 |