Best Known (252−41, 252, s)-Nets in Base 2
(252−41, 252, 520)-Net over F2 — Constructive and digital
Digital (211, 252, 520)-net over F2, using
- 22 times duplication [i] based on digital (209, 250, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 50, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 50, 104)-net over F32, using
(252−41, 252, 1382)-Net over F2 — Digital
Digital (211, 252, 1382)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 1382, F2, 2, 41) (dual of [(1382, 2), 2512, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2252, 2066, F2, 2, 41) (dual of [(2066, 2), 3880, 42]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2248, 2064, F2, 2, 41) (dual of [(2064, 2), 3880, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2248, 4128, F2, 41) (dual of [4128, 3880, 42]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2247, 4127, F2, 41) (dual of [4127, 3880, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,18]) [i] based on
- linear OA(2241, 4097, F2, 41) (dual of [4097, 3856, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2217, 4097, F2, 37) (dual of [4097, 3880, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-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(2247, 4127, F2, 41) (dual of [4127, 3880, 42]-code), using
- OOA 2-folding [i] based on linear OA(2248, 4128, F2, 41) (dual of [4128, 3880, 42]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2248, 2064, F2, 2, 41) (dual of [(2064, 2), 3880, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2252, 2066, F2, 2, 41) (dual of [(2066, 2), 3880, 42]-NRT-code), using
(252−41, 252, 49770)-Net in Base 2 — Upper bound on s
There is no (211, 252, 49771)-net in base 2, because
- 1 times m-reduction [i] would yield (211, 251, 49771)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3618 662309 511039 846453 474034 249569 671140 303662 245806 100798 036846 154617 886116 > 2251 [i]