Best Known (31, 31+23, s)-Nets in Base 256
(31, 31+23, 5960)-Net over F256 — Constructive and digital
Digital (31, 54, 5960)-net over F256, using
- 2561 times duplication [i] based on digital (30, 53, 5960)-net over F256, using
- net defined by OOA [i] based on linear OOA(25653, 5960, F256, 23, 23) (dual of [(5960, 23), 137027, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25653, 65561, F256, 23) (dual of [65561, 65508, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25653, 65562, F256, 23) (dual of [65562, 65509, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2568, 26, F256, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,256)), using
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- Reed–Solomon code RS(248,256) [i]
- discarding factors / shortening the dual code based on linear OA(2568, 256, F256, 8) (dual of [256, 248, 9]-code or 256-arc in PG(7,256)), using
- construction X applied to Ce(22) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(25653, 65562, F256, 23) (dual of [65562, 65509, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(25653, 65561, F256, 23) (dual of [65561, 65508, 24]-code), using
- net defined by OOA [i] based on linear OOA(25653, 5960, F256, 23, 23) (dual of [(5960, 23), 137027, 24]-NRT-code), using
(31, 31+23, 40718)-Net over F256 — Digital
Digital (31, 54, 40718)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(25654, 40718, F256, 23) (dual of [40718, 40664, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(25654, 65566, F256, 23) (dual of [65566, 65512, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
- 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(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,11]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25654, 65566, F256, 23) (dual of [65566, 65512, 24]-code), using
(31, 31+23, large)-Net in Base 256 — Upper bound on s
There is no (31, 54, large)-net in base 256, because
- 21 times m-reduction [i] would yield (31, 33, large)-net in base 256, but