Best Known (58, 58+11, s)-Nets in Base 25
(58, 58+11, 1873037)-Net over F25 — Constructive and digital
Digital (58, 69, 1873037)-net over F25, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(2518, 195317, F25, 5, 5) (dual of [(195317, 5), 976567, 6]-NRT-code), using
- digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (13, 18, 195317)-net over F25, using
(58, 58+11, large)-Net over F25 — Digital
Digital (58, 69, large)-net over F25, using
- t-expansion [i] based on digital (57, 69, large)-net over F25, using
- 3 times m-reduction [i] based on digital (57, 72, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2572, large, F25, 15) (dual of [large, large−72, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2572, large, F25, 15) (dual of [large, large−72, 16]-code), using
- 3 times m-reduction [i] based on digital (57, 72, large)-net over F25, using
(58, 58+11, large)-Net in Base 25 — Upper bound on s
There is no (58, 69, large)-net in base 25, because
- 9 times m-reduction [i] would yield (58, 60, large)-net in base 25, but