Best Known (59−27, 59, s)-Nets in Base 128
(59−27, 59, 1261)-Net over F128 — Constructive and digital
Digital (32, 59, 1261)-net over F128, using
- 1283 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
(59−27, 59, 7790)-Net over F128 — Digital
Digital (32, 59, 7790)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12859, 7790, F128, 2, 27) (dual of [(7790, 2), 15521, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12859, 8202, F128, 2, 27) (dual of [(8202, 2), 16345, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12859, 16404, F128, 27) (dual of [16404, 16345, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(19) [i] based on
- linear OA(12853, 16384, F128, 27) (dual of [16384, 16331, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(26) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(12859, 16404, F128, 27) (dual of [16404, 16345, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(12859, 8202, F128, 2, 27) (dual of [(8202, 2), 16345, 28]-NRT-code), using
(59−27, 59, large)-Net in Base 128 — Upper bound on s
There is no (32, 59, large)-net in base 128, because
- 25 times m-reduction [i] would yield (32, 34, large)-net in base 128, but