Best Known (37, 37+35, s)-Nets in Base 128
(37, 37+35, 964)-Net over F128 — Constructive and digital
Digital (37, 72, 964)-net over F128, using
- 1282 times duplication [i] based on digital (35, 70, 964)-net over F128, using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(12865, 16385, F128, 33) (dual of [16385, 16320, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12870, 16390, F128, 35) (dual of [16390, 16320, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(12870, 16389, F128, 35) (dual of [16389, 16319, 36]-code), using
- net defined by OOA [i] based on linear OOA(12870, 964, F128, 35, 35) (dual of [(964, 35), 33670, 36]-NRT-code), using
(37, 37+35, 4777)-Net over F128 — Digital
Digital (37, 72, 4777)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12872, 4777, F128, 3, 35) (dual of [(4777, 3), 14259, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 5465, F128, 3, 35) (dual of [(5465, 3), 16323, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12872, 16395, F128, 35) (dual of [16395, 16323, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12872, 16396, F128, 35) (dual of [16396, 16324, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,15]) [i] based on
- linear OA(12869, 16385, F128, 35) (dual of [16385, 16316, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- 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(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([0,17]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12872, 16396, F128, 35) (dual of [16396, 16324, 36]-code), using
- OOA 3-folding [i] based on linear OA(12872, 16395, F128, 35) (dual of [16395, 16323, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(12872, 5465, F128, 3, 35) (dual of [(5465, 3), 16323, 36]-NRT-code), using
(37, 37+35, large)-Net in Base 128 — Upper bound on s
There is no (37, 72, large)-net in base 128, because
- 33 times m-reduction [i] would yield (37, 39, large)-net in base 128, but