Best Known (223−40, 223, s)-Nets in Base 2
(223−40, 223, 320)-Net over F2 — Constructive and digital
Digital (183, 223, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (183, 225, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 45, 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, 45, 64)-net over F32, using
(223−40, 223, 868)-Net over F2 — Digital
Digital (183, 223, 868)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2223, 868, F2, 2, 40) (dual of [(868, 2), 1513, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 1030, F2, 2, 40) (dual of [(1030, 2), 1837, 41]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2222, 1030, F2, 2, 40) (dual of [(1030, 2), 1838, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2222, 2060, F2, 40) (dual of [2060, 1838, 41]-code), using
- strength reduction [i] based on linear OA(2222, 2060, F2, 41) (dual of [2060, 1838, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(40) ⊂ Ce(38) [i] based on
- strength reduction [i] based on linear OA(2222, 2060, F2, 41) (dual of [2060, 1838, 42]-code), using
- OOA 2-folding [i] based on linear OA(2222, 2060, F2, 40) (dual of [2060, 1838, 41]-code), using
- 21 times duplication [i] based on linear OOA(2222, 1030, F2, 2, 40) (dual of [(1030, 2), 1838, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2223, 1030, F2, 2, 40) (dual of [(1030, 2), 1837, 41]-NRT-code), using
(223−40, 223, 18841)-Net in Base 2 — Upper bound on s
There is no (183, 223, 18842)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 13 489447 014990 705632 607539 257917 104826 308114 980023 870095 022824 540042 > 2223 [i]