Best Known (126, 126+28, s)-Nets in Base 4
(126, 126+28, 1223)-Net over F4 — Constructive and digital
Digital (126, 154, 1223)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (28, 42, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 65)-net over F64, using
- digital (84, 112, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- digital (28, 42, 195)-net over F4, using
(126, 126+28, 12256)-Net over F4 — Digital
Digital (126, 154, 12256)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4154, 12256, F4, 28) (dual of [12256, 12102, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4154, 16412, F4, 28) (dual of [16412, 16258, 29]-code), using
- 1 times truncation [i] based on linear OA(4155, 16413, F4, 29) (dual of [16413, 16258, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- 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(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4120, 16384, F4, 23) (dual of [16384, 16264, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(4155, 16413, F4, 29) (dual of [16413, 16258, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4154, 16412, F4, 28) (dual of [16412, 16258, 29]-code), using
(126, 126+28, large)-Net in Base 4 — Upper bound on s
There is no (126, 154, large)-net in base 4, because
- 26 times m-reduction [i] would yield (126, 128, large)-net in base 4, but