Best Known (204, 249, s)-Nets in Base 2
(204, 249, 320)-Net over F2 — Constructive and digital
Digital (204, 249, 320)-net over F2, using
- t-expansion [i] based on digital (203, 249, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (203, 250, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 50, 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, 50, 64)-net over F32, using
- 1 times m-reduction [i] based on digital (203, 250, 320)-net over F2, using
(204, 249, 913)-Net over F2 — Digital
Digital (204, 249, 913)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2249, 913, F2, 2, 45) (dual of [(913, 2), 1577, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2249, 1038, F2, 2, 45) (dual of [(1038, 2), 1827, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2249, 2076, F2, 45) (dual of [2076, 1827, 46]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2249, 2077, F2, 45) (dual of [2077, 1828, 46]-code), using
- OOA 2-folding [i] based on linear OA(2249, 2076, F2, 45) (dual of [2076, 1827, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(2249, 1038, F2, 2, 45) (dual of [(1038, 2), 1827, 46]-NRT-code), using
(204, 249, 22369)-Net in Base 2 — Upper bound on s
There is no (204, 249, 22370)-net in base 2, because
- 1 times m-reduction [i] would yield (204, 248, 22370)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 452 633745 181238 429937 517976 240544 617834 876406 975433 629571 936258 407251 838192 > 2248 [i]