Best Known (51, 51+25, s)-Nets in Base 128
(51, 51+25, 174763)-Net over F128 — Constructive and digital
Digital (51, 76, 174763)-net over F128, using
- 1282 times duplication [i] based on digital (49, 74, 174763)-net over F128, using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12867, 2097153, F128, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12874, 2097160, F128, 25) (dual of [2097160, 2097086, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(12874, 2097157, F128, 25) (dual of [2097157, 2097083, 26]-code), using
- net defined by OOA [i] based on linear OOA(12874, 174763, F128, 25, 25) (dual of [(174763, 25), 4369001, 26]-NRT-code), using
(51, 51+25, 872835)-Net over F128 — Digital
Digital (51, 76, 872835)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12876, 872835, F128, 2, 25) (dual of [(872835, 2), 1745594, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12876, 1048584, F128, 2, 25) (dual of [(1048584, 2), 2097092, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12876, 2097168, F128, 25) (dual of [2097168, 2097092, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(12873, 2097153, F128, 25) (dual of [2097153, 2097080, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(12861, 2097153, F128, 21) (dual of [2097153, 2097092, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(12876, 2097168, F128, 25) (dual of [2097168, 2097092, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(12876, 1048584, F128, 2, 25) (dual of [(1048584, 2), 2097092, 26]-NRT-code), using
(51, 51+25, large)-Net in Base 128 — Upper bound on s
There is no (51, 76, large)-net in base 128, because
- 23 times m-reduction [i] would yield (51, 53, large)-net in base 128, but