Best Known (244−36, 244, s)-Nets in Base 2
(244−36, 244, 624)-Net over F2 — Constructive and digital
Digital (208, 244, 624)-net over F2, using
- 2 times m-reduction [i] based on digital (208, 246, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 41, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 41, 104)-net over F64, using
(244−36, 244, 2319)-Net over F2 — Digital
Digital (208, 244, 2319)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 2319, F2, 3, 36) (dual of [(2319, 3), 6713, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 2742, F2, 3, 36) (dual of [(2742, 3), 7982, 37]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2242, 2742, F2, 3, 36) (dual of [(2742, 3), 7984, 37]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2242, 8226, F2, 36) (dual of [8226, 7984, 37]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2240, 8224, F2, 36) (dual of [8224, 7984, 37]-code), using
- 1 times truncation [i] based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2235, 8193, F2, 37) (dual of [8193, 7958, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2209, 8193, F2, 33) (dual of [8193, 7984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 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([0,18]) ⊂ C([0,16]) [i] based on
- 1 times truncation [i] based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2240, 8224, F2, 36) (dual of [8224, 7984, 37]-code), using
- OOA 3-folding [i] based on linear OA(2242, 8226, F2, 36) (dual of [8226, 7984, 37]-code), using
- 22 times duplication [i] based on linear OOA(2242, 2742, F2, 3, 36) (dual of [(2742, 3), 7984, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 2742, F2, 3, 36) (dual of [(2742, 3), 7982, 37]-NRT-code), using
(244−36, 244, 90914)-Net in Base 2 — Upper bound on s
There is no (208, 244, 90915)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 28 273313 973960 679464 016788 157778 414848 867295 273648 910910 142550 279843 098158 > 2244 [i]