Best Known (250−45, 250, s)-Nets in Base 2
(250−45, 250, 380)-Net over F2 — Constructive and digital
Digital (205, 250, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 50, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
(250−45, 250, 929)-Net over F2 — Digital
Digital (205, 250, 929)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 929, F2, 2, 45) (dual of [(929, 2), 1608, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2250, 1039, F2, 2, 45) (dual of [(1039, 2), 1828, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2250, 2078, F2, 45) (dual of [2078, 1828, 46]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2249, 2077, F2, 45) (dual of [2077, 1828, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,20]) [i] based on
- linear OA(2243, 2049, F2, 45) (dual of [2049, 1806, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- 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(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([0,22]) ⊂ C([0,20]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2249, 2077, F2, 45) (dual of [2077, 1828, 46]-code), using
- OOA 2-folding [i] based on linear OA(2250, 2078, F2, 45) (dual of [2078, 1828, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(2250, 1039, F2, 2, 45) (dual of [(1039, 2), 1828, 46]-NRT-code), using
(250−45, 250, 23086)-Net in Base 2 — Upper bound on s
There is no (205, 250, 23087)-net in base 2, because
- 1 times m-reduction [i] would yield (205, 249, 23087)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 905 212246 545997 551053 381844 934327 946010 565734 083613 272789 962271 513728 272102 > 2249 [i]