Best Known (31, 31+15, s)-Nets in Base 128
(31, 31+15, 299595)-Net over F128 — Constructive and digital
Digital (31, 46, 299595)-net over F128, using
- net defined by OOA [i] based on linear OOA(12846, 299595, F128, 15, 15) (dual of [(299595, 15), 4493879, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12846, 2097166, F128, 15) (dual of [2097166, 2097120, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12846, 2097166, F128, 15) (dual of [2097166, 2097120, 16]-code), using
(31, 31+15, 1048584)-Net over F128 — Digital
Digital (31, 46, 1048584)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12846, 1048584, F128, 2, 15) (dual of [(1048584, 2), 2097122, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12831, 2097153, F128, 11) (dual of [2097153, 2097122, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,7]) ⊂ C([0,5]) [i] based on
- OOA 2-folding [i] based on linear OA(12846, 2097168, F128, 15) (dual of [2097168, 2097122, 16]-code), using
(31, 31+15, large)-Net in Base 128 — Upper bound on s
There is no (31, 46, large)-net in base 128, because
- 13 times m-reduction [i] would yield (31, 33, large)-net in base 128, but