Best Known (143, 143+18, s)-Nets in Base 2
(143, 143+18, 14568)-Net over F2 — Constructive and digital
Digital (143, 161, 14568)-net over F2, using
- t-expansion [i] based on digital (142, 161, 14568)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 14568, F2, 19, 19) (dual of [(14568, 19), 276631, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2161, 131113, F2, 19) (dual of [131113, 130952, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2161, 131113, F2, 19) (dual of [131113, 130952, 20]-code), using
- net defined by OOA [i] based on linear OOA(2161, 14568, F2, 19, 19) (dual of [(14568, 19), 276631, 20]-NRT-code), using
(143, 143+18, 26222)-Net over F2 — Digital
Digital (143, 161, 26222)-net over F2, using
- 21 times duplication [i] based on digital (142, 160, 26222)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2160, 26222, F2, 5, 18) (dual of [(26222, 5), 130950, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2160, 131110, F2, 18) (dual of [131110, 130950, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 131112, F2, 18) (dual of [131112, 130952, 19]-code), using
- 1 times truncation [i] based on linear OA(2161, 131113, F2, 19) (dual of [131113, 130952, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2161, 131113, F2, 19) (dual of [131113, 130952, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 131112, F2, 18) (dual of [131112, 130952, 19]-code), using
- OOA 5-folding [i] based on linear OA(2160, 131110, F2, 18) (dual of [131110, 130950, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2160, 26222, F2, 5, 18) (dual of [(26222, 5), 130950, 19]-NRT-code), using
(143, 143+18, 1006556)-Net in Base 2 — Upper bound on s
There is no (143, 161, 1006557)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 923026 878992 153999 019388 325687 434625 405950 283654 > 2161 [i]