Best Known (17, 17+15, s)-Nets in Base 256
(17, 17+15, 9363)-Net over F256 — Constructive and digital
Digital (17, 32, 9363)-net over F256, using
- 2562 times duplication [i] based on digital (15, 30, 9363)-net over F256, using
- net defined by OOA [i] based on linear OOA(25630, 9363, F256, 15, 15) (dual of [(9363, 15), 140415, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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(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([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(25630, 65542, F256, 15) (dual of [65542, 65512, 16]-code), using
- net defined by OOA [i] based on linear OOA(25630, 9363, F256, 15, 15) (dual of [(9363, 15), 140415, 16]-NRT-code), using
(17, 17+15, 21849)-Net over F256 — Digital
Digital (17, 32, 21849)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25632, 21849, F256, 3, 15) (dual of [(21849, 3), 65515, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25632, 65547, F256, 15) (dual of [65547, 65515, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25632, 65548, F256, 15) (dual of [65548, 65516, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25632, 65548, F256, 15) (dual of [65548, 65516, 16]-code), using
- OOA 3-folding [i] based on linear OA(25632, 65547, F256, 15) (dual of [65547, 65515, 16]-code), using
(17, 17+15, large)-Net in Base 256 — Upper bound on s
There is no (17, 32, large)-net in base 256, because
- 13 times m-reduction [i] would yield (17, 19, large)-net in base 256, but