Best Known (78, 78+22, s)-Nets in Base 4
(78, 78+22, 1040)-Net over F4 — Constructive and digital
Digital (78, 100, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 25, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
(78, 78+22, 2629)-Net over F4 — Digital
Digital (78, 100, 2629)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4100, 2629, F4, 22) (dual of [2629, 2529, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 4111, F4, 22) (dual of [4111, 4011, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(43, 15, F4, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 4111, F4, 22) (dual of [4111, 4011, 23]-code), using
(78, 78+22, 486583)-Net in Base 4 — Upper bound on s
There is no (78, 100, 486584)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 606938 432145 905033 348831 685148 113028 364764 557882 423944 346870 > 4100 [i]