Best Known (231−40, 231, s)-Nets in Base 2
(231−40, 231, 380)-Net over F2 — Constructive and digital
Digital (191, 231, 380)-net over F2, using
- 21 times duplication [i] based on digital (190, 230, 380)-net over F2, using
- t-expansion [i] based on digital (189, 230, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 46, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 46, 76)-net over F32, using
- t-expansion [i] based on digital (189, 230, 380)-net over F2, using
(231−40, 231, 1016)-Net over F2 — Digital
Digital (191, 231, 1016)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 1016, F2, 2, 40) (dual of [(1016, 2), 1801, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2231, 1042, F2, 2, 40) (dual of [(1042, 2), 1853, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2231, 2084, F2, 40) (dual of [2084, 1853, 41]-code), using
- 1 times truncation [i] based on linear OA(2232, 2085, F2, 41) (dual of [2085, 1853, 42]-code), using
- construction XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [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(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 5, F2, 1) (dual of [5, 4, 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 XX applied to Ce(40) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- 1 times truncation [i] based on linear OA(2232, 2085, F2, 41) (dual of [2085, 1853, 42]-code), using
- OOA 2-folding [i] based on linear OA(2231, 2084, F2, 40) (dual of [2084, 1853, 41]-code), using
- discarding factors / shortening the dual code based on linear OOA(2231, 1042, F2, 2, 40) (dual of [(1042, 2), 1853, 41]-NRT-code), using
(231−40, 231, 24870)-Net in Base 2 — Upper bound on s
There is no (191, 231, 24871)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3451 685247 490224 133478 431867 408218 288417 105273 340397 048043 654166 055746 > 2231 [i]