Best Known (156−27, 156, s)-Nets in Base 4
(156−27, 156, 1340)-Net over F4 — Constructive and digital
Digital (129, 156, 1340)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (35, 48, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F64, using
- digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- digital (35, 48, 312)-net over F4, using
(156−27, 156, 1415)-Net in Base 4 — Constructive
(129, 156, 1415)-net in base 4, using
- (u, u+v)-construction [i] based on
- (35, 48, 387)-net in base 4, using
- trace code for nets [i] based on (3, 16, 129)-net in base 64, using
- 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- trace code for nets [i] based on (3, 16, 129)-net in base 64, using
- digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- (35, 48, 387)-net in base 4, using
(156−27, 156, 16443)-Net over F4 — Digital
Digital (129, 156, 16443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4156, 16443, F4, 27) (dual of [16443, 16287, 28]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4155, 16441, F4, 27) (dual of [16441, 16286, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- 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(499, 16385, F4, 19) (dual of [16385, 16286, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [i] based on
- linear OA(4155, 16442, F4, 26) (dual of [16442, 16287, 27]-code), using Gilbert–Varšamov bound and bm = 4155 > Vbs−1(k−1) = 134 240445 470095 403677 994777 843457 811180 218729 071063 270399 071730 372963 130593 570484 047791 420912 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(4155, 16441, F4, 27) (dual of [16441, 16286, 28]-code), using
- construction X with Varšamov bound [i] based on
(156−27, 156, large)-Net in Base 4 — Upper bound on s
There is no (129, 156, large)-net in base 4, because
- 25 times m-reduction [i] would yield (129, 131, large)-net in base 4, but