Best Known (136, 164, s)-Nets in Base 2
(136, 164, 320)-Net over F2 — Constructive and digital
Digital (136, 164, 320)-net over F2, using
- t-expansion [i] based on digital (135, 164, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (135, 165, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 33, 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, 33, 64)-net over F32, using
- 1 times m-reduction [i] based on digital (135, 165, 320)-net over F2, using
(136, 164, 873)-Net over F2 — Digital
Digital (136, 164, 873)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2164, 873, F2, 2, 28) (dual of [(873, 2), 1582, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2164, 1040, F2, 2, 28) (dual of [(1040, 2), 1916, 29]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2160, 1038, F2, 2, 28) (dual of [(1038, 2), 1916, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2160, 2076, F2, 28) (dual of [2076, 1916, 29]-code), using
- 1 times truncation [i] based on linear OA(2161, 2077, F2, 29) (dual of [2077, 1916, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2155, 2049, F2, 29) (dual of [2049, 1894, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2133, 2049, F2, 25) (dual of [2049, 1916, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 1 times truncation [i] based on linear OA(2161, 2077, F2, 29) (dual of [2077, 1916, 30]-code), using
- OOA 2-folding [i] based on linear OA(2160, 2076, F2, 28) (dual of [2076, 1916, 29]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2160, 1038, F2, 2, 28) (dual of [(1038, 2), 1916, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2164, 1040, F2, 2, 28) (dual of [(1040, 2), 1916, 29]-NRT-code), using
(136, 164, 20294)-Net in Base 2 — Upper bound on s
There is no (136, 164, 20295)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 23 397748 377076 285334 734103 686803 508908 981032 309544 > 2164 [i]