Best Known (72, 72+20, s)-Nets in Base 4
(72, 72+20, 1040)-Net over F4 — Constructive and digital
Digital (72, 92, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 23, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
(72, 72+20, 2771)-Net over F4 — Digital
Digital (72, 92, 2771)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(492, 2771, F4, 20) (dual of [2771, 2679, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(492, 4109, F4, 20) (dual of [4109, 4017, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(479, 4096, F4, 18) (dual of [4096, 4017, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(492, 4109, F4, 20) (dual of [4109, 4017, 21]-code), using
(72, 72+20, 522156)-Net in Base 4 — Upper bound on s
There is no (72, 92, 522157)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 24 520296 126329 131848 973554 705306 730719 939493 556899 189450 > 492 [i]