Best Known (57−25, 57, s)-Nets in Base 256
(57−25, 57, 5463)-Net over F256 — Constructive and digital
Digital (32, 57, 5463)-net over F256, using
- 2561 times duplication [i] based on digital (31, 56, 5463)-net over F256, using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- 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(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(2567, 23, F256, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,256)), using
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- Reed–Solomon code RS(249,256) [i]
- discarding factors / shortening the dual code based on linear OA(2567, 256, F256, 7) (dual of [256, 249, 8]-code or 256-arc in PG(6,256)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25656, 65560, F256, 25) (dual of [65560, 65504, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25656, 65557, F256, 25) (dual of [65557, 65501, 26]-code), using
- net defined by OOA [i] based on linear OOA(25656, 5463, F256, 25, 25) (dual of [(5463, 25), 136519, 26]-NRT-code), using
(57−25, 57, 32781)-Net over F256 — Digital
Digital (32, 57, 32781)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25657, 32781, F256, 2, 25) (dual of [(32781, 2), 65505, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25657, 65562, F256, 25) (dual of [65562, 65505, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(15) [i] based on
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(24) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(25657, 65562, F256, 25) (dual of [65562, 65505, 26]-code), using
(57−25, 57, large)-Net in Base 256 — Upper bound on s
There is no (32, 57, large)-net in base 256, because
- 23 times m-reduction [i] would yield (32, 34, large)-net in base 256, but