Best Known (177−32, 177, s)-Nets in Base 4
(177−32, 177, 1223)-Net over F4 — Constructive and digital
Digital (145, 177, 1223)-net over F4, using
- 41 times duplication [i] based on digital (144, 176, 1223)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (32, 48, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 16, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 16, 65)-net over F64, using
- digital (96, 128, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 257)-net over F256, using
- digital (32, 48, 195)-net over F4, using
- (u, u+v)-construction [i] based on
(177−32, 177, 13646)-Net over F4 — Digital
Digital (145, 177, 13646)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4177, 13646, F4, 32) (dual of [13646, 13469, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 16421, F4, 32) (dual of [16421, 16244, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4176, 16420, F4, 32) (dual of [16420, 16244, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4141, 16385, F4, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4176, 16420, F4, 32) (dual of [16420, 16244, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 16421, F4, 32) (dual of [16421, 16244, 33]-code), using
(177−32, 177, large)-Net in Base 4 — Upper bound on s
There is no (145, 177, large)-net in base 4, because
- 30 times m-reduction [i] would yield (145, 147, large)-net in base 4, but