Best Known (61, 61+19, s)-Nets in Base 16
(61, 61+19, 14566)-Net over F16 — Constructive and digital
Digital (61, 80, 14566)-net over F16, using
- net defined by OOA [i] based on linear OOA(1680, 14566, F16, 19, 19) (dual of [(14566, 19), 276674, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1680, 131095, F16, 19) (dual of [131095, 131015, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 131096, F16, 19) (dual of [131096, 131016, 20]-code), using
- trace code [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(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(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([0,9]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1680, 131096, F16, 19) (dual of [131096, 131016, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1680, 131095, F16, 19) (dual of [131095, 131015, 20]-code), using
(61, 61+19, 131096)-Net over F16 — Digital
Digital (61, 80, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1680, 131096, F16, 19) (dual of [131096, 131016, 20]-code), using
- trace code [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [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(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(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([0,9]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
(61, 61+19, large)-Net in Base 16 — Upper bound on s
There is no (61, 80, large)-net in base 16, because
- 17 times m-reduction [i] would yield (61, 63, large)-net in base 16, but