Best Known (148, 148+34, s)-Nets in Base 4
(148, 148+34, 1158)-Net over F4 — Constructive and digital
Digital (148, 182, 1158)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (29, 46, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 23, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 23, 65)-net over F16, using
- digital (102, 136, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 34, 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, 34, 257)-net over F256, using
- digital (29, 46, 130)-net over F4, using
(148, 148+34, 10818)-Net over F4 — Digital
Digital (148, 182, 10818)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4182, 10818, F4, 34) (dual of [10818, 10636, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4182, 16412, F4, 34) (dual of [16412, 16230, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(33) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4182, 16412, F4, 34) (dual of [16412, 16230, 35]-code), using
(148, 148+34, 6674311)-Net in Base 4 — Upper bound on s
There is no (148, 182, 6674312)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 37 576688 901051 144864 052910 943864 115359 026387 686391 692229 858415 473309 876281 914929 711523 271864 004067 491079 128680 > 4182 [i]