Best Known (13, 13+7, s)-Nets in Base 25
(13, 13+7, 5211)-Net over F25 — Constructive and digital
Digital (13, 20, 5211)-net over F25, using
- net defined by OOA [i] based on linear OOA(2520, 5211, F25, 7, 7) (dual of [(5211, 7), 36457, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2520, 15634, F25, 7) (dual of [15634, 15614, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(2513, 15626, F25, 5) (dual of [15626, 15613, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(257, 8, F25, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,25)), using
- dual of repetition code with length 8 [i]
- linear OA(251, 8, F25, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- Reed–Solomon code RS(24,25) [i]
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2520, 15634, F25, 7) (dual of [15634, 15614, 8]-code), using
(13, 13+7, 15634)-Net over F25 — Digital
Digital (13, 20, 15634)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2520, 15634, F25, 7) (dual of [15634, 15614, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(2513, 15626, F25, 5) (dual of [15626, 15613, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 256−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(257, 8, F25, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,25)), using
- dual of repetition code with length 8 [i]
- linear OA(251, 8, F25, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- Reed–Solomon code RS(24,25) [i]
- discarding factors / shortening the dual code based on linear OA(251, 25, F25, 1) (dual of [25, 24, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
(13, 13+7, large)-Net in Base 25 — Upper bound on s
There is no (13, 20, large)-net in base 25, because
- 5 times m-reduction [i] would yield (13, 15, large)-net in base 25, but