Best Known (157−26, 157, s)-Nets in Base 2
(157−26, 157, 390)-Net over F2 — Constructive and digital
Digital (131, 157, 390)-net over F2, using
- 21 times duplication [i] based on digital (130, 156, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 26, 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, 26, 65)-net over F64, using
(157−26, 157, 1127)-Net over F2 — Digital
Digital (131, 157, 1127)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 1127, F2, 3, 26) (dual of [(1127, 3), 3224, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 1369, F2, 3, 26) (dual of [(1369, 3), 3950, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2157, 4107, F2, 26) (dual of [4107, 3950, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2157, 4108, F2, 26) (dual of [4108, 3951, 27]-code), using
- 1 times truncation [i] based on linear OA(2158, 4109, F2, 27) (dual of [4109, 3951, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2145, 4096, F2, 25) (dual of [4096, 3951, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2158, 4109, F2, 27) (dual of [4109, 3951, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2157, 4108, F2, 26) (dual of [4108, 3951, 27]-code), using
- OOA 3-folding [i] based on linear OA(2157, 4107, F2, 26) (dual of [4107, 3950, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 1369, F2, 3, 26) (dual of [(1369, 3), 3950, 27]-NRT-code), using
(157−26, 157, 24467)-Net in Base 2 — Upper bound on s
There is no (131, 157, 24468)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182759 813227 735216 557416 234098 757222 537943 211572 > 2157 [i]