Best Known (126−27, 126, s)-Nets in Base 4
(126−27, 126, 1045)-Net over F4 — Constructive and digital
Digital (99, 126, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, 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 (5, 18, 17)-net over F4, using
(126−27, 126, 3454)-Net over F4 — Digital
Digital (99, 126, 3454)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4126, 3454, F4, 27) (dual of [3454, 3328, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 4119, F4, 27) (dual of [4119, 3993, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-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
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 4119, F4, 27) (dual of [4119, 3993, 28]-code), using
(126−27, 126, 1162303)-Net in Base 4 — Upper bound on s
There is no (99, 126, 1162304)-net in base 4, because
- 1 times m-reduction [i] would yield (99, 125, 1162304)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1809 267197 704530 061915 579697 895963 972755 050771 051118 846961 244732 876981 745497 > 4125 [i]