Best Known (68−29, 68, s)-Nets in Base 256
(68−29, 68, 4683)-Net over F256 — Constructive and digital
Digital (39, 68, 4683)-net over F256, using
- 2562 times duplication [i] based on digital (37, 66, 4683)-net over F256, using
- net defined by OOA [i] based on linear OOA(25666, 4683, F256, 29, 29) (dual of [(4683, 29), 135741, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25666, 65563, F256, 29) (dual of [65563, 65497, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(25666, 65566, F256, 29) (dual of [65566, 65500, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2569, 29, F256, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,256)), using
- discarding factors / shortening the dual code based on linear OA(2569, 256, F256, 9) (dual of [256, 247, 10]-code or 256-arc in PG(8,256)), using
- Reed–Solomon code RS(247,256) [i]
- discarding factors / shortening the dual code based on linear OA(2569, 256, F256, 9) (dual of [256, 247, 10]-code or 256-arc in PG(8,256)), using
- construction X applied to C([0,14]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25666, 65566, F256, 29) (dual of [65566, 65500, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25666, 65563, F256, 29) (dual of [65563, 65497, 30]-code), using
- net defined by OOA [i] based on linear OOA(25666, 4683, F256, 29, 29) (dual of [(4683, 29), 135741, 30]-NRT-code), using
(68−29, 68, 40526)-Net over F256 — Digital
Digital (39, 68, 40526)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25668, 40526, F256, 29) (dual of [40526, 40458, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(25668, 65572, F256, 29) (dual of [65572, 65504, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,8]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(25611, 35, F256, 11) (dual of [35, 24, 12]-code or 35-arc in PG(10,256)), using
- discarding factors / shortening the dual code based on linear OA(25611, 256, F256, 11) (dual of [256, 245, 12]-code or 256-arc in PG(10,256)), using
- Reed–Solomon code RS(245,256) [i]
- discarding factors / shortening the dual code based on linear OA(25611, 256, F256, 11) (dual of [256, 245, 12]-code or 256-arc in PG(10,256)), using
- construction X applied to C([0,14]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25668, 65572, F256, 29) (dual of [65572, 65504, 30]-code), using
(68−29, 68, large)-Net in Base 256 — Upper bound on s
There is no (39, 68, large)-net in base 256, because
- 27 times m-reduction [i] would yield (39, 41, large)-net in base 256, but