Best Known (31, 31+7, s)-Nets in Base 25
(31, 31+7, 3188075)-Net over F25 — Constructive and digital
Digital (31, 38, 3188075)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 7, 391875)-net over F25, using
- net defined by OOA [i] based on linear OOA(257, 391875, F25, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(257, 391875, F25, 2, 3) (dual of [(391875, 2), 783743, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(257, 391875, F25, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
- digital (24, 31, 2796200)-net over F25, using
- net defined by OOA [i] based on linear OOA(2531, 2796200, F25, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2531, 8388601, F25, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2531, large, F25, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2531, large, F25, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2531, 8388601, F25, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2531, 2796200, F25, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- digital (4, 7, 391875)-net over F25, using
(31, 31+7, large)-Net over F25 — Digital
Digital (31, 38, large)-net over F25, using
- 252 times duplication [i] based on digital (29, 36, large)-net over F25, using
- t-expansion [i] based on digital (28, 36, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- t-expansion [i] based on digital (28, 36, large)-net over F25, using
(31, 31+7, large)-Net in Base 25 — Upper bound on s
There is no (31, 38, large)-net in base 25, because
- 5 times m-reduction [i] would yield (31, 33, large)-net in base 25, but