Best Known (157−21, 157, s)-Nets in Base 2
(157−21, 157, 3280)-Net over F2 — Constructive and digital
Digital (136, 157, 3280)-net over F2, using
- net defined by OOA [i] based on linear OOA(2157, 3280, F2, 21, 21) (dual of [(3280, 21), 68723, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2151, 32769, F2, 21) (dual of [32769, 32618, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2121, 32769, F2, 17) (dual of [32769, 32648, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
(157−21, 157, 6560)-Net over F2 — Digital
Digital (136, 157, 6560)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 6560, F2, 5, 21) (dual of [(6560, 5), 32643, 22]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2157, 32800, F2, 21) (dual of [32800, 32643, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2151, 32769, F2, 21) (dual of [32769, 32618, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2121, 32769, F2, 17) (dual of [32769, 32648, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 230−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2157, 32801, F2, 21) (dual of [32801, 32644, 22]-code), using
- OOA 5-folding [i] based on linear OA(2157, 32800, F2, 21) (dual of [32800, 32643, 22]-code), using
(157−21, 157, 224913)-Net in Base 2 — Upper bound on s
There is no (136, 157, 224914)-net in base 2, because
- 1 times m-reduction [i] would yield (136, 156, 224914)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 91344 379392 059966 100722 908369 388085 016872 996328 > 2156 [i]