Best Known (13, 13+5, s)-Nets in Base 25
(13, 13+5, 195317)-Net over F25 — Constructive and digital
Digital (13, 18, 195317)-net over F25, using
- net defined by OOA [i] based on linear OOA(2518, 195317, F25, 5, 5) (dual of [(195317, 5), 976567, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2518, 390635, F25, 5) (dual of [390635, 390617, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2517, 390626, F25, 5) (dual of [390626, 390609, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(259, 390626, F25, 3) (dual of [390626, 390617, 4]-code or 390626-cap in PG(8,25)), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(251, 9, F25, 1) (dual of [9, 8, 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([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2518, 390635, F25, 5) (dual of [390635, 390617, 6]-code), using
(13, 13+5, 390636)-Net over F25 — Digital
Digital (13, 18, 390636)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2518, 390636, F25, 5) (dual of [390636, 390618, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(2517, 390626, F25, 5) (dual of [390626, 390609, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(259, 390626, F25, 3) (dual of [390626, 390617, 4]-code or 390626-cap in PG(8,25)), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(259, 10, F25, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,25)), using
- dual of repetition code with length 10 [i]
- linear OA(251, 10, F25, 1) (dual of [10, 9, 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([0,2]) ⊂ C([0,1]) [i] based on
(13, 13+5, large)-Net in Base 25 — Upper bound on s
There is no (13, 18, large)-net in base 25, because
- 3 times m-reduction [i] would yield (13, 15, large)-net in base 25, but