Best Known (68−33, 68, s)-Nets in Base 256
(68−33, 68, 4096)-Net over F256 — Constructive and digital
Digital (35, 68, 4096)-net over F256, using
- 2563 times duplication [i] based on digital (32, 65, 4096)-net over F256, using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
(68−33, 68, 14145)-Net over F256 — Digital
Digital (35, 68, 14145)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25668, 14145, F256, 4, 33) (dual of [(14145, 4), 56512, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25668, 16387, F256, 4, 33) (dual of [(16387, 4), 65480, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25668, 65548, F256, 33) (dual of [65548, 65480, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 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(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- OOA 4-folding [i] based on linear OA(25668, 65548, F256, 33) (dual of [65548, 65480, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(25668, 16387, F256, 4, 33) (dual of [(16387, 4), 65480, 34]-NRT-code), using
(68−33, 68, large)-Net in Base 256 — Upper bound on s
There is no (35, 68, large)-net in base 256, because
- 31 times m-reduction [i] would yield (35, 37, large)-net in base 256, but