Best Known (121−29, 121, s)-Nets in Base 16
(121−29, 121, 9364)-Net over F16 — Constructive and digital
Digital (92, 121, 9364)-net over F16, using
- net defined by OOA [i] based on linear OOA(16121, 9364, F16, 29, 29) (dual of [(9364, 29), 271435, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16121, 131097, F16, 29) (dual of [131097, 130976, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16120, 131096, F16, 29) (dual of [131096, 130976, 30]-code), using
- trace code [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- trace code [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16120, 131096, F16, 29) (dual of [131096, 130976, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16121, 131097, F16, 29) (dual of [131097, 130976, 30]-code), using
(121−29, 121, 131098)-Net over F16 — Digital
Digital (92, 121, 131098)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16121, 131098, F16, 29) (dual of [131098, 130977, 30]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16120, 131096, F16, 29) (dual of [131096, 130976, 30]-code), using
- trace code [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- trace code [i] based on linear OA(25660, 65548, F256, 29) (dual of [65548, 65488, 30]-code), using
- linear OA(16120, 131097, F16, 28) (dual of [131097, 130977, 29]-code), using Gilbert–Varšamov bound and bm = 16120 > Vbs−1(k−1) = 7784 651885 655532 981864 611762 878339 419243 065541 574371 900499 763001 629281 347965 083897 763486 976549 752121 137879 641266 039807 169820 520029 460230 946816 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16120, 131096, F16, 29) (dual of [131096, 130976, 30]-code), using
- construction X with Varšamov bound [i] based on
(121−29, 121, large)-Net in Base 16 — Upper bound on s
There is no (92, 121, large)-net in base 16, because
- 27 times m-reduction [i] would yield (92, 94, large)-net in base 16, but