Best Known (51, 51+17, s)-Nets in Base 25
(51, 51+17, 48830)-Net over F25 — Constructive and digital
Digital (51, 68, 48830)-net over F25, using
- net defined by OOA [i] based on linear OOA(2568, 48830, F25, 17, 17) (dual of [(48830, 17), 830042, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2568, 390641, F25, 17) (dual of [390641, 390573, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 390645, F25, 17) (dual of [390645, 390577, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2565, 390626, F25, 17) (dual of [390626, 390561, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2549, 390626, F25, 13) (dual of [390626, 390577, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(253, 19, F25, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,25) or 19-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2568, 390645, F25, 17) (dual of [390645, 390577, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2568, 390641, F25, 17) (dual of [390641, 390573, 18]-code), using
(51, 51+17, 390645)-Net over F25 — Digital
Digital (51, 68, 390645)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2568, 390645, F25, 17) (dual of [390645, 390577, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(2565, 390626, F25, 17) (dual of [390626, 390561, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2549, 390626, F25, 13) (dual of [390626, 390577, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(253, 19, F25, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,25) or 19-cap in PG(2,25)), using
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- Reed–Solomon code RS(22,25) [i]
- discarding factors / shortening the dual code based on linear OA(253, 25, F25, 3) (dual of [25, 22, 4]-code or 25-arc in PG(2,25) or 25-cap in PG(2,25)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
(51, 51+17, large)-Net in Base 25 — Upper bound on s
There is no (51, 68, large)-net in base 25, because
- 15 times m-reduction [i] would yield (51, 53, large)-net in base 25, but