Best Known (64−29, 64, s)-Nets in Base 128
(64−29, 64, 1171)-Net over F128 — Constructive and digital
Digital (35, 64, 1171)-net over F128, using
- 1284 times duplication [i] based on digital (31, 60, 1171)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 1171, F128, 29, 29) (dual of [(1171, 29), 33899, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12860, 16395, F128, 29) (dual of [16395, 16335, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 16396, F128, 29) (dual of [16396, 16336, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(12857, 16385, F128, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(12849, 16385, F128, 25) (dual of [16385, 16336, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 16396, F128, 29) (dual of [16396, 16336, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(12860, 16395, F128, 29) (dual of [16395, 16335, 30]-code), using
- net defined by OOA [i] based on linear OOA(12860, 1171, F128, 29, 29) (dual of [(1171, 29), 33899, 30]-NRT-code), using
(64−29, 64, 8204)-Net over F128 — Digital
Digital (35, 64, 8204)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12864, 8204, F128, 2, 29) (dual of [(8204, 2), 16344, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12864, 16408, F128, 29) (dual of [16408, 16344, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,10]) [i] based on
- linear OA(12857, 16385, F128, 29) (dual of [16385, 16328, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,14]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(12864, 16408, F128, 29) (dual of [16408, 16344, 30]-code), using
(64−29, 64, large)-Net in Base 128 — Upper bound on s
There is no (35, 64, large)-net in base 128, because
- 27 times m-reduction [i] would yield (35, 37, large)-net in base 128, but