Best Known (181−30, 181, s)-Nets in Base 2
(181−30, 181, 390)-Net over F2 — Constructive and digital
Digital (151, 181, 390)-net over F2, using
- 21 times duplication [i] based on digital (150, 180, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 65)-net over F64, using
(181−30, 181, 1176)-Net over F2 — Digital
Digital (151, 181, 1176)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2181, 1176, F2, 3, 30) (dual of [(1176, 3), 3347, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2181, 1369, F2, 3, 30) (dual of [(1369, 3), 3926, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2181, 4107, F2, 30) (dual of [4107, 3926, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 4108, F2, 30) (dual of [4108, 3927, 31]-code), using
- 1 times truncation [i] based on linear OA(2182, 4109, F2, 31) (dual of [4109, 3927, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2169, 4096, F2, 29) (dual of [4096, 3927, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2182, 4109, F2, 31) (dual of [4109, 3927, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 4108, F2, 30) (dual of [4108, 3927, 31]-code), using
- OOA 3-folding [i] based on linear OA(2181, 4107, F2, 30) (dual of [4107, 3926, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2181, 1369, F2, 3, 30) (dual of [(1369, 3), 3926, 31]-NRT-code), using
(181−30, 181, 27532)-Net in Base 2 — Upper bound on s
There is no (151, 181, 27533)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 065524 125305 203262 620916 628559 726051 085050 335470 424760 > 2181 [i]