Best Known (67−31, 67, s)-Nets in Base 128
(67−31, 67, 1093)-Net over F128 — Constructive and digital
Digital (36, 67, 1093)-net over F128, using
- 1283 times duplication [i] based on digital (33, 64, 1093)-net over F128, using
- net defined by OOA [i] based on linear OOA(12864, 1093, F128, 31, 31) (dual of [(1093, 31), 33819, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(12861, 16385, F128, 31) (dual of [16385, 16324, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- 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(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([0,15]) ⊂ C([0,13]) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(12864, 16396, F128, 31) (dual of [16396, 16332, 32]-code), using
- net defined by OOA [i] based on linear OOA(12864, 1093, F128, 31, 31) (dual of [(1093, 31), 33819, 32]-NRT-code), using
(67−31, 67, 6920)-Net over F128 — Digital
Digital (36, 67, 6920)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12867, 6920, F128, 2, 31) (dual of [(6920, 2), 13773, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12867, 8202, F128, 2, 31) (dual of [(8202, 2), 16337, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12867, 16404, F128, 31) (dual of [16404, 16337, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(12847, 16384, F128, 24) (dual of [16384, 16337, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(30) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(12867, 16404, F128, 31) (dual of [16404, 16337, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(12867, 8202, F128, 2, 31) (dual of [(8202, 2), 16337, 32]-NRT-code), using
(67−31, 67, large)-Net in Base 128 — Upper bound on s
There is no (36, 67, large)-net in base 128, because
- 29 times m-reduction [i] would yield (36, 38, large)-net in base 128, but