Best Known (226−41, 226, s)-Nets in Base 2
(226−41, 226, 320)-Net over F2 — Constructive and digital
Digital (185, 226, 320)-net over F2, using
- 21 times duplication [i] based on digital (184, 225, 320)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (183, 225, 320)-net over F2, using
(226−41, 226, 840)-Net over F2 — Digital
Digital (185, 226, 840)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2226, 840, F2, 2, 41) (dual of [(840, 2), 1454, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 1032, F2, 2, 41) (dual of [(1032, 2), 1838, 42]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2222, 1030, F2, 2, 41) (dual of [(1030, 2), 1838, 42]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(2222, 2060, F2, 41) (dual of [2060, 1838, 42]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2222, 1030, F2, 2, 41) (dual of [(1030, 2), 1838, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 1032, F2, 2, 41) (dual of [(1032, 2), 1838, 42]-NRT-code), using
(226−41, 226, 20195)-Net in Base 2 — Upper bound on s
There is no (185, 226, 20196)-net in base 2, because
- 1 times m-reduction [i] would yield (185, 225, 20196)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 53 932717 453734 474010 939750 869412 928046 603844 272782 549201 463614 116456 > 2225 [i]