Best Known (131, 131+13, s)-Nets in Base 2
(131, 131+13, 1398100)-Net over F2 — Constructive and digital
Digital (131, 144, 1398100)-net over F2, using
- 25 times duplication [i] based on digital (126, 139, 1398100)-net over F2, using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2139, 8388601, F2, 13) (dual of [8388601, 8388462, 14]-code), using
- net defined by OOA [i] based on linear OOA(2139, 1398100, F2, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
(131, 131+13, 1677721)-Net over F2 — Digital
Digital (131, 144, 1677721)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2144, 1677721, F2, 5, 13) (dual of [(1677721, 5), 8388461, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2139, 1677720, F2, 5, 13) (dual of [(1677720, 5), 8388461, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2139, 8388600, F2, 13) (dual of [8388600, 8388461, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8388609 | 246−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2139, large, F2, 13) (dual of [large, large−139, 14]-code), using
- OOA 5-folding [i] based on linear OA(2139, 8388600, F2, 13) (dual of [8388600, 8388461, 14]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2139, 1677720, F2, 5, 13) (dual of [(1677720, 5), 8388461, 14]-NRT-code), using
(131, 131+13, large)-Net in Base 2 — Upper bound on s
There is no (131, 144, large)-net in base 2, because
- 11 times m-reduction [i] would yield (131, 133, large)-net in base 2, but