Best Known (60−27, 60, s)-Nets in Base 128
(60−27, 60, 1262)-Net over F128 — Constructive and digital
Digital (33, 60, 1262)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 1262, F128, 27, 27) (dual of [(1262, 27), 34014, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12860, 16407, F128, 27) (dual of [16407, 16347, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [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(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,13]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(12860, 16407, F128, 27) (dual of [16407, 16347, 28]-code), using
(60−27, 60, 8204)-Net over F128 — Digital
Digital (33, 60, 8204)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12860, 8204, F128, 2, 27) (dual of [(8204, 2), 16348, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,9]) [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(12837, 16385, F128, 19) (dual of [16385, 16348, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,13]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(12860, 16408, F128, 27) (dual of [16408, 16348, 28]-code), using
(60−27, 60, large)-Net in Base 128 — Upper bound on s
There is no (33, 60, large)-net in base 128, because
- 25 times m-reduction [i] would yield (33, 35, large)-net in base 128, but