Best Known (231−29, 231, s)-Nets in Base 2
(231−29, 231, 4683)-Net over F2 — Constructive and digital
Digital (202, 231, 4683)-net over F2, using
- net defined by OOA [i] based on linear OOA(2231, 4683, F2, 29, 29) (dual of [(4683, 29), 135576, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2231, 65563, F2, 29) (dual of [65563, 65332, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2231, 65563, F2, 29) (dual of [65563, 65332, 30]-code), using
(231−29, 231, 10822)-Net over F2 — Digital
Digital (202, 231, 10822)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2231, 10822, F2, 6, 29) (dual of [(10822, 6), 64701, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2231, 10928, F2, 6, 29) (dual of [(10928, 6), 65337, 30]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2231, 65568, F2, 29) (dual of [65568, 65337, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2225, 65537, F2, 29) (dual of [65537, 65312, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2231, 65569, F2, 29) (dual of [65569, 65338, 30]-code), using
- OOA 6-folding [i] based on linear OA(2231, 65568, F2, 29) (dual of [65568, 65337, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2231, 10928, F2, 6, 29) (dual of [(10928, 6), 65337, 30]-NRT-code), using
(231−29, 231, 533253)-Net in Base 2 — Upper bound on s
There is no (202, 231, 533254)-net in base 2, because
- 1 times m-reduction [i] would yield (202, 230, 533254)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1725 452803 311554 828591 810733 707380 733033 580818 494259 214969 744117 260928 > 2230 [i]