Best Known (60−29, 60, s)-Nets in Base 256
(60−29, 60, 4681)-Net over F256 — Constructive and digital
Digital (31, 60, 4681)-net over F256, using
- 2563 times duplication [i] based on digital (28, 57, 4681)-net over F256, using
- net defined by OOA [i] based on linear OOA(25657, 4681, F256, 29, 29) (dual of [(4681, 29), 135692, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25657, 65535, F256, 29) (dual of [65535, 65478, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(25657, 65536, F256, 29) (dual of [65536, 65479, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(25657, 65535, F256, 29) (dual of [65535, 65478, 30]-code), using
- net defined by OOA [i] based on linear OOA(25657, 4681, F256, 29, 29) (dual of [(4681, 29), 135692, 30]-NRT-code), using
(60−29, 60, 15986)-Net over F256 — Digital
Digital (31, 60, 15986)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25660, 15986, F256, 4, 29) (dual of [(15986, 4), 63884, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25660, 16387, F256, 4, 29) (dual of [(16387, 4), 65488, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [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(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- OOA 4-folding [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(25660, 16387, F256, 4, 29) (dual of [(16387, 4), 65488, 30]-NRT-code), using
(60−29, 60, large)-Net in Base 256 — Upper bound on s
There is no (31, 60, large)-net in base 256, because
- 27 times m-reduction [i] would yield (31, 33, large)-net in base 256, but