Best Known (39−19, 39, s)-Nets in Base 256
(39−19, 39, 7282)-Net over F256 — Constructive and digital
Digital (20, 39, 7282)-net over F256, using
- 2561 times duplication [i] based on digital (19, 38, 7282)-net over F256, using
- net defined by OOA [i] based on linear OOA(25638, 7282, F256, 19, 19) (dual of [(7282, 19), 138320, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25638, 65539, F256, 19) (dual of [65539, 65501, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25638, 65539, F256, 19) (dual of [65539, 65501, 20]-code), using
- net defined by OOA [i] based on linear OOA(25638, 7282, F256, 19, 19) (dual of [(7282, 19), 138320, 20]-NRT-code), using
(39−19, 39, 16386)-Net over F256 — Digital
Digital (20, 39, 16386)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25639, 16386, F256, 4, 19) (dual of [(16386, 4), 65505, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25639, 65544, F256, 19) (dual of [65544, 65505, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(25637, 65536, F256, 19) (dual of [65536, 65499, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 4-folding [i] based on linear OA(25639, 65544, F256, 19) (dual of [65544, 65505, 20]-code), using
(39−19, 39, large)-Net in Base 256 — Upper bound on s
There is no (20, 39, large)-net in base 256, because
- 17 times m-reduction [i] would yield (20, 22, large)-net in base 256, but