Best Known (208, 208+41, s)-Nets in Base 2
(208, 208+41, 490)-Net over F2 — Constructive and digital
Digital (208, 249, 490)-net over F2, using
- t-expansion [i] based on digital (207, 249, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (207, 250, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 50, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 50, 98)-net over F32, using
- 1 times m-reduction [i] based on digital (207, 250, 490)-net over F2, using
(208, 208+41, 1376)-Net over F2 — Digital
Digital (208, 249, 1376)-net over F2, using
- 21 times duplication [i] based on digital (207, 248, 1376)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 1376, F2, 3, 41) (dual of [(1376, 3), 3880, 42]-NRT-code), using
- OOA 3-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 3-folding [i] based on linear OA(2248, 4128, F2, 41) (dual of [4128, 3880, 42]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 1376, F2, 3, 41) (dual of [(1376, 3), 3880, 42]-NRT-code), using
(208, 208+41, 44853)-Net in Base 2 — Upper bound on s
There is no (208, 249, 44854)-net in base 2, because
- 1 times m-reduction [i] would yield (208, 248, 44854)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 452 488960 180957 829511 739790 817552 857434 037124 844804 853092 706969 611614 087331 > 2248 [i]