Best Known (49, 49+15, s)-Nets in Base 16
(49, 49+15, 18727)-Net over F16 — Constructive and digital
Digital (49, 64, 18727)-net over F16, using
- net defined by OOA [i] based on linear OOA(1664, 18727, F16, 15, 15) (dual of [(18727, 15), 280841, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1664, 131090, F16, 15) (dual of [131090, 131026, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131096, F16, 15) (dual of [131096, 131032, 16]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(25632, 65548, F256, 15) (dual of [65548, 65516, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(1664, 131096, F16, 15) (dual of [131096, 131032, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(1664, 131090, F16, 15) (dual of [131090, 131026, 16]-code), using
(49, 49+15, 131096)-Net over F16 — Digital
Digital (49, 64, 131096)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1664, 131096, F16, 15) (dual of [131096, 131032, 16]-code), using
- trace code [i] 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
- trace code [i] based on linear OA(25632, 65548, F256, 15) (dual of [65548, 65516, 16]-code), using
(49, 49+15, large)-Net in Base 16 — Upper bound on s
There is no (49, 64, large)-net in base 16, because
- 13 times m-reduction [i] would yield (49, 51, large)-net in base 16, but