Best Known (58−27, 58, s)-Nets in Base 128
(58−27, 58, 1261)-Net over F128 — Constructive and digital
Digital (31, 58, 1261)-net over F128, using
- 1282 times duplication [i] based on digital (29, 56, 1261)-net over F128, using
- net defined by OOA [i] based on linear OOA(12856, 1261, F128, 27, 27) (dual of [(1261, 27), 33991, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12856, 16394, F128, 27) (dual of [16394, 16338, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1283, 11, F128, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12856, 16396, F128, 27) (dual of [16396, 16340, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12856, 16394, F128, 27) (dual of [16394, 16338, 28]-code), using
- net defined by OOA [i] based on linear OOA(12856, 1261, F128, 27, 27) (dual of [(1261, 27), 33991, 28]-NRT-code), using
(58−27, 58, 6362)-Net over F128 — Digital
Digital (31, 58, 6362)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12858, 6362, F128, 2, 27) (dual of [(6362, 2), 12666, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12858, 8201, F128, 2, 27) (dual of [(8201, 2), 16344, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12858, 16402, F128, 27) (dual of [16402, 16344, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(12853, 16385, F128, 27) (dual of [16385, 16332, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(12841, 16385, F128, 21) (dual of [16385, 16344, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(12858, 16402, F128, 27) (dual of [16402, 16344, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(12858, 8201, F128, 2, 27) (dual of [(8201, 2), 16344, 28]-NRT-code), using
(58−27, 58, large)-Net in Base 128 — Upper bound on s
There is no (31, 58, large)-net in base 128, because
- 25 times m-reduction [i] would yield (31, 33, large)-net in base 128, but