Best Known (97−15, 97, s)-Nets in Base 25
(97−15, 97, 1328582)-Net over F25 — Constructive and digital
Digital (82, 97, 1328582)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (19, 26, 130211)-net over F25, using
- net defined by OOA [i] based on linear OOA(2526, 130211, F25, 7, 7) (dual of [(130211, 7), 911451, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2526, 390634, F25, 7) (dual of [390634, 390608, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2526, 390635, F25, 7) (dual of [390635, 390609, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(2525, 390626, F25, 7) (dual of [390626, 390601, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 390626 | 258−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 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(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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2526, 390635, F25, 7) (dual of [390635, 390609, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2526, 390634, F25, 7) (dual of [390634, 390608, 8]-code), using
- net defined by OOA [i] based on linear OOA(2526, 130211, F25, 7, 7) (dual of [(130211, 7), 911451, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F25, using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2571, large, F25, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2571, 8388598, F25, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2571, 1198371, F25, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (19, 26, 130211)-net over F25, using
(97−15, 97, large)-Net over F25 — Digital
Digital (82, 97, large)-net over F25, using
- t-expansion [i] based on digital (81, 97, large)-net over F25, using
- 5 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(25101, large, F25, 21) (dual of [large, large−101, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25102, large, F25, 21) (dual of [large, large−102, 22]-code), using
- 5 times m-reduction [i] based on digital (81, 102, large)-net over F25, using
(97−15, 97, large)-Net in Base 25 — Upper bound on s
There is no (82, 97, large)-net in base 25, because
- 13 times m-reduction [i] would yield (82, 84, large)-net in base 25, but