Best Known (26−9, 26, s)-Nets in Base 25
(26−9, 26, 3908)-Net over F25 — Constructive and digital
Digital (17, 26, 3908)-net over F25, using
- net defined by OOA [i] based on linear OOA(2526, 3908, F25, 9, 9) (dual of [(3908, 9), 35146, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2526, 15633, F25, 9) (dual of [15633, 15607, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2525, 15626, F25, 9) (dual of [15626, 15601, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2519, 15626, F25, 7) (dual of [15626, 15607, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(2526, 15633, F25, 9) (dual of [15633, 15607, 10]-code), using
(26−9, 26, 13844)-Net over F25 — Digital
Digital (17, 26, 13844)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2526, 13844, F25, 9) (dual of [13844, 13818, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2526, 15633, F25, 9) (dual of [15633, 15607, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(2525, 15626, F25, 9) (dual of [15626, 15601, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(2519, 15626, F25, 7) (dual of [15626, 15607, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(251, 7, F25, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2526, 15633, F25, 9) (dual of [15633, 15607, 10]-code), using
(26−9, 26, large)-Net in Base 25 — Upper bound on s
There is no (17, 26, large)-net in base 25, because
- 7 times m-reduction [i] would yield (17, 19, large)-net in base 25, but