Best Known (159−26, 159, s)-Nets in Base 2
(159−26, 159, 390)-Net over F2 — Constructive and digital
Digital (133, 159, 390)-net over F2, using
- 23 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
(159−26, 159, 1203)-Net over F2 — Digital
Digital (133, 159, 1203)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2159, 1203, F2, 3, 26) (dual of [(1203, 3), 3450, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 1370, F2, 3, 26) (dual of [(1370, 3), 3951, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2159, 4110, F2, 26) (dual of [4110, 3951, 27]-code), using
- 2 times code embedding in larger space [i] 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
- 2 times code embedding in larger space [i] based on linear OA(2157, 4108, F2, 26) (dual of [4108, 3951, 27]-code), using
- OOA 3-folding [i] based on linear OA(2159, 4110, F2, 26) (dual of [4110, 3951, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(2159, 1370, F2, 3, 26) (dual of [(1370, 3), 3951, 27]-NRT-code), using
(159−26, 159, 27222)-Net in Base 2 — Upper bound on s
There is no (133, 159, 27223)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 730845 026361 681772 396493 245494 619381 166264 261128 > 2159 [i]