Best Known (228−41, 228, s)-Nets in Base 2
(228−41, 228, 320)-Net over F2 — Constructive and digital
Digital (187, 228, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (187, 230, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 46, 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, 46, 64)-net over F32, using
(228−41, 228, 873)-Net over F2 — Digital
Digital (187, 228, 873)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 873, F2, 2, 41) (dual of [(873, 2), 1518, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2228, 1039, F2, 2, 41) (dual of [(1039, 2), 1850, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2228, 2078, F2, 41) (dual of [2078, 1850, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2227, 2077, F2, 41) (dual of [2077, 1850, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(2221, 2049, F2, 41) (dual of [2049, 1828, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2199, 2049, F2, 37) (dual of [2049, 1850, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (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([0,20]) ⊂ C([0,18]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2227, 2077, F2, 41) (dual of [2077, 1850, 42]-code), using
- OOA 2-folding [i] based on linear OA(2228, 2078, F2, 41) (dual of [2078, 1850, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(2228, 1039, F2, 2, 41) (dual of [(1039, 2), 1850, 42]-NRT-code), using
(228−41, 228, 21647)-Net in Base 2 — Upper bound on s
There is no (187, 228, 21648)-net in base 2, because
- 1 times m-reduction [i] would yield (187, 227, 21648)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 215 800256 043396 812848 110240 498585 684063 913372 978363 728315 345351 089896 > 2227 [i]