Best Known (244−37, 244, s)-Nets in Base 2
(244−37, 244, 624)-Net over F2 — Constructive and digital
Digital (207, 244, 624)-net over F2, using
- 24 times duplication [i] based on digital (203, 240, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 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, 40, 104)-net over F64, using
(244−37, 244, 2057)-Net over F2 — Digital
Digital (207, 244, 2057)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 2057, F2, 4, 37) (dual of [(2057, 4), 7984, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2244, 8228, F2, 37) (dual of [8228, 7984, 38]-code), using
- 3 times code embedding in larger space [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
- 3 times code embedding in larger space [i] based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- OOA 4-folding [i] based on linear OA(2244, 8228, F2, 37) (dual of [8228, 7984, 38]-code), using
(244−37, 244, 87478)-Net in Base 2 — Upper bound on s
There is no (207, 244, 87479)-net in base 2, because
- 1 times m-reduction [i] would yield (207, 243, 87479)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14 134972 431961 519042 796746 571531 700276 139556 383605 646349 299682 365669 569529 > 2243 [i]