Best Known (130, 130+27, s)-Nets in Base 4
(130, 130+27, 1340)-Net over F4 — Constructive and digital
Digital (130, 157, 1340)-net over F4, using
- 41 times duplication [i] based on 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
- (u, u+v)-construction [i] based on
(130, 130+27, 1415)-Net in Base 4 — Constructive
(130, 157, 1415)-net in base 4, using
- 41 times duplication [i] based on (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
- (u, u+v)-construction [i] based on
(130, 130+27, 16445)-Net over F4 — Digital
Digital (130, 157, 16445)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 16445, F4, 27) (dual of [16445, 16288, 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, 16443, F4, 26) (dual of [16443, 16288, 27]-code), using Gilbert–Varšamov bound and bm = 4155 > Vbs−1(k−1) = 134 444864 235052 168053 708160 129687 976636 264203 356704 717389 555951 611310 085523 167911 162508 013805 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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
(130, 130+27, large)-Net in Base 4 — Upper bound on s
There is no (130, 157, large)-net in base 4, because
- 25 times m-reduction [i] would yield (130, 132, large)-net in base 4, but