Best Known (161−28, 161, s)-Nets in Base 2
(161−28, 161, 320)-Net over F2 — Constructive and digital
Digital (133, 161, 320)-net over F2, using
- 21 times duplication [i] based on digital (132, 160, 320)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (131, 160, 320)-net over F2, using
(161−28, 161, 800)-Net over F2 — Digital
Digital (133, 161, 800)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 800, F2, 2, 28) (dual of [(800, 2), 1439, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2161, 1038, F2, 2, 28) (dual of [(1038, 2), 1915, 29]-NRT-code), using
- strength reduction [i] based on linear OOA(2161, 1038, F2, 2, 29) (dual of [(1038, 2), 1915, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2161, 2076, F2, 29) (dual of [2076, 1915, 30]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2161, 2077, F2, 29) (dual of [2077, 1916, 30]-code), using
- OOA 2-folding [i] based on linear OA(2161, 2076, F2, 29) (dual of [2076, 1915, 30]-code), using
- strength reduction [i] based on linear OOA(2161, 1038, F2, 2, 29) (dual of [(1038, 2), 1915, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2161, 1038, F2, 2, 28) (dual of [(1038, 2), 1915, 29]-NRT-code), using
(161−28, 161, 17490)-Net in Base 2 — Upper bound on s
There is no (133, 161, 17491)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 924925 167041 220671 191107 791784 434323 400337 307924 > 2161 [i]