Best Known (31, 31+27, s)-Nets in Base 256
(31, 31+27, 5042)-Net over F256 — Constructive and digital
Digital (31, 58, 5042)-net over F256, using
- 2562 times duplication [i] based on digital (29, 56, 5042)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(25656, 65547, F256, 27) (dual of [65547, 65491, 28]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5042, F256, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
(31, 31+27, 21210)-Net over F256 — Digital
Digital (31, 58, 21210)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25658, 21210, F256, 3, 27) (dual of [(21210, 3), 63572, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25658, 21851, F256, 3, 27) (dual of [(21851, 3), 65495, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25658, 65553, F256, 27) (dual of [65553, 65495, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(25658, 65554, F256, 27) (dual of [65554, 65496, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(25653, 65537, F256, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25658, 65554, F256, 27) (dual of [65554, 65496, 28]-code), using
- OOA 3-folding [i] based on linear OA(25658, 65553, F256, 27) (dual of [65553, 65495, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(25658, 21851, F256, 3, 27) (dual of [(21851, 3), 65495, 28]-NRT-code), using
(31, 31+27, large)-Net in Base 256 — Upper bound on s
There is no (31, 58, large)-net in base 256, because
- 25 times m-reduction [i] would yield (31, 33, large)-net in base 256, but