Best Known (239−40, 239, s)-Nets in Base 2
(239−40, 239, 490)-Net over F2 — Constructive and digital
Digital (199, 239, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (199, 240, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 98)-net over F32, using
(239−40, 239, 1103)-Net over F2 — Digital
Digital (199, 239, 1103)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2239, 1103, F2, 40) (dual of [1103, 864, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 2111, F2, 40) (dual of [2111, 1872, 41]-code), using
- 1 times truncation [i] based on linear OA(2240, 2112, F2, 41) (dual of [2112, 1872, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- 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(2177, 2049, F2, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(219, 63, F2, 7) (dual of [63, 44, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- 1 times truncation [i] based on linear OA(2240, 2112, F2, 41) (dual of [2112, 1872, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 2111, F2, 40) (dual of [2111, 1872, 41]-code), using
(239−40, 239, 32826)-Net in Base 2 — Upper bound on s
There is no (199, 239, 32827)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 883686 398932 550212 703576 073311 148143 486305 502636 131608 081450 909130 342666 > 2239 [i]