Best Known (26−13, 26, s)-Nets in Base 128
(26−13, 26, 2731)-Net over F128 — Constructive and digital
Digital (13, 26, 2731)-net over F128, using
- net defined by OOA [i] based on linear OOA(12826, 2731, F128, 13, 13) (dual of [(2731, 13), 35477, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(12826, 16387, F128, 13) (dual of [16387, 16361, 14]-code), using
(26−13, 26, 5463)-Net over F128 — Digital
Digital (13, 26, 5463)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12826, 5463, F128, 3, 13) (dual of [(5463, 3), 16363, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12826, 16389, F128, 13) (dual of [16389, 16363, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(12821, 16385, F128, 11) (dual of [16385, 16364, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12826, 16390, F128, 13) (dual of [16390, 16364, 14]-code), using
- OOA 3-folding [i] based on linear OA(12826, 16389, F128, 13) (dual of [16389, 16363, 14]-code), using
(26−13, 26, large)-Net in Base 128 — Upper bound on s
There is no (13, 26, large)-net in base 128, because
- 11 times m-reduction [i] would yield (13, 15, large)-net in base 128, but