Best Known (59−28, 59, s)-Nets in Base 128
(59−28, 59, 1171)-Net over F128 — Constructive and digital
Digital (31, 59, 1171)-net over F128, using
- 1 times m-reduction [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
(59−28, 59, 5466)-Net over F128 — Digital
Digital (31, 59, 5466)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12859, 5466, F128, 3, 28) (dual of [(5466, 3), 16339, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12859, 16398, F128, 28) (dual of [16398, 16339, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1284, 14, F128, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(12859, 16398, F128, 28) (dual of [16398, 16339, 29]-code), using
(59−28, 59, large)-Net in Base 128 — Upper bound on s
There is no (31, 59, large)-net in base 128, because
- 26 times m-reduction [i] would yield (31, 33, large)-net in base 128, but