Best Known (80−19, 80, s)-Nets in Base 25
(80−19, 80, 43406)-Net over F25 — Constructive and digital
Digital (61, 80, 43406)-net over F25, using
- 251 times duplication [i] based on digital (60, 79, 43406)-net over F25, using
- net defined by OOA [i] based on linear OOA(2579, 43406, F25, 19, 19) (dual of [(43406, 19), 824635, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2579, 390655, F25, 19) (dual of [390655, 390576, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(2573, 390626, F25, 19) (dual of [390626, 390553, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(256, 29, F25, 5) (dual of [29, 23, 6]-code), using
- construction for [s, s−6, 6]-code from [s, 3, s−3]-code [i] based on linear OA(2526, 29, F25, 25) (dual of [29, 3, 26]-code), using
- construction X applied to C1 ⊂ C0 [i] based on
- linear OA(2525, 26, F25, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,25)), using code C1 for u = 2 by de Boer and Brouwer [i]
- linear OA(2523, 26, F25, 23) (dual of [26, 3, 24]-code or 26-arc in PG(22,25)), using code C0 for u = 2 by de Boer and Brouwer [i]
- linear OA(251, 3, F25, 1) (dual of [3, 2, 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 C1 ⊂ C0 [i] based on
- construction for [s, s−6, 6]-code from [s, 3, s−3]-code [i] based on linear OA(2526, 29, F25, 25) (dual of [29, 3, 26]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2579, 390655, F25, 19) (dual of [390655, 390576, 20]-code), using
- net defined by OOA [i] based on linear OOA(2579, 43406, F25, 19, 19) (dual of [(43406, 19), 824635, 20]-NRT-code), using
(80−19, 80, 514035)-Net over F25 — Digital
Digital (61, 80, 514035)-net over F25, using
(80−19, 80, large)-Net in Base 25 — Upper bound on s
There is no (61, 80, large)-net in base 25, because
- 17 times m-reduction [i] would yield (61, 63, large)-net in base 25, but