Best Known (115, 115+25, s)-Nets in Base 2
(115, 115+25, 320)-Net over F2 — Constructive and digital
Digital (115, 140, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 28, 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
(115, 115+25, 692)-Net over F2 — Digital
Digital (115, 140, 692)-net over F2, using
- 21 times duplication [i] based on digital (114, 139, 692)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2139, 692, F2, 3, 25) (dual of [(692, 3), 1937, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2139, 2076, F2, 25) (dual of [2076, 1937, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, 2077, F2, 25) (dual of [2077, 1938, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 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(2111, 2049, F2, 21) (dual of [2049, 1938, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2139, 2077, F2, 25) (dual of [2077, 1938, 26]-code), using
- OOA 3-folding [i] based on linear OA(2139, 2076, F2, 25) (dual of [2076, 1937, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2139, 692, F2, 3, 25) (dual of [(692, 3), 1937, 26]-NRT-code), using
(115, 115+25, 16211)-Net in Base 2 — Upper bound on s
There is no (115, 140, 16212)-net in base 2, because
- 1 times m-reduction [i] would yield (115, 139, 16212)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 697143 192413 016275 485249 526454 042302 523204 > 2139 [i]