Best Known (181−33, 181, s)-Nets in Base 4
(181−33, 181, 1223)-Net over F4 — Constructive and digital
Digital (148, 181, 1223)-net over F4, using
- 41 times duplication [i] based on digital (147, 180, 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 (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- digital (32, 48, 195)-net over F4, using
- (u, u+v)-construction [i] based on
(181−33, 181, 12939)-Net over F4 — Digital
Digital (148, 181, 12939)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 12939, F4, 33) (dual of [12939, 12758, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 16425, F4, 33) (dual of [16425, 16244, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4179, 16423, F4, 33) (dual of [16423, 16244, 34]-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(410, 38, F4, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4179, 16423, F4, 33) (dual of [16423, 16244, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 16425, F4, 33) (dual of [16425, 16244, 34]-code), using
(181−33, 181, large)-Net in Base 4 — Upper bound on s
There is no (148, 181, large)-net in base 4, because
- 31 times m-reduction [i] would yield (148, 150, large)-net in base 4, but