Best Known (76−15, 76, s)-Nets in Base 4
(76−15, 76, 1076)-Net over F4 — Constructive and digital
Digital (61, 76, 1076)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 8, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- trace code for nets [i] based on digital (1, 8, 24)-net over F16, using
- digital (45, 60, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 15, 257)-net over F256, using
- digital (9, 16, 48)-net over F4, using
(76−15, 76, 4317)-Net over F4 — Digital
Digital (61, 76, 4317)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(476, 4317, F4, 15) (dual of [4317, 4241, 16]-code), using
- 206 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 27 times 0, 1, 53 times 0, 1, 98 times 0) [i] based on linear OA(467, 4102, F4, 15) (dual of [4102, 4035, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- 206 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 27 times 0, 1, 53 times 0, 1, 98 times 0) [i] based on linear OA(467, 4102, F4, 15) (dual of [4102, 4035, 16]-code), using
(76−15, 76, 3180090)-Net in Base 4 — Upper bound on s
There is no (61, 76, 3180091)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 75, 3180091)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1427 250419 039525 771233 613800 803830 010501 883648 > 475 [i]