Best Known (159−27, 159, s)-Nets in Base 2
(159−27, 159, 320)-Net over F2 — Constructive and digital
Digital (132, 159, 320)-net over F2, using
- t-expansion [i] based on digital (131, 159, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (131, 160, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 32, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 32, 64)-net over F32, using
- 1 times m-reduction [i] based on digital (131, 160, 320)-net over F2, using
(159−27, 159, 1027)-Net over F2 — Digital
Digital (132, 159, 1027)-net over F2, using
- 21 times duplication [i] based on digital (131, 158, 1027)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 1027, F2, 4, 27) (dual of [(1027, 4), 3950, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2158, 4108, F2, 27) (dual of [4108, 3950, 28]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2158, 4109, F2, 27) (dual of [4109, 3951, 28]-code), using
- OOA 4-folding [i] based on linear OA(2158, 4108, F2, 27) (dual of [4108, 3950, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2158, 1027, F2, 4, 27) (dual of [(1027, 4), 3950, 28]-NRT-code), using
(159−27, 159, 25808)-Net in Base 2 — Upper bound on s
There is no (132, 159, 25809)-net in base 2, because
- 1 times m-reduction [i] would yield (132, 158, 25809)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 365509 286174 015019 054275 804666 133830 150978 070504 > 2158 [i]