Best Known (85, 85+27, s)-Nets in Base 16
(85, 85+27, 10084)-Net over F16 — Constructive and digital
Digital (85, 112, 10084)-net over F16, using
- net defined by OOA [i] based on linear OOA(16112, 10084, F16, 27, 27) (dual of [(10084, 27), 272156, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16112, 131093, F16, 27) (dual of [131093, 130981, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16112, 131096, F16, 27) (dual of [131096, 130984, 28]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(16112, 131096, F16, 27) (dual of [131096, 130984, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(16112, 131093, F16, 27) (dual of [131093, 130981, 28]-code), using
(85, 85+27, 131096)-Net over F16 — Digital
Digital (85, 112, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16112, 131096, F16, 27) (dual of [131096, 130984, 28]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(25656, 65548, F256, 27) (dual of [65548, 65492, 28]-code), using
(85, 85+27, large)-Net in Base 16 — Upper bound on s
There is no (85, 112, large)-net in base 16, because
- 25 times m-reduction [i] would yield (85, 87, large)-net in base 16, but