Best Known (51, 51+3, s)-Nets in Base 25
(51, 51+3, large)-Net over F25 — Constructive and digital
Digital (51, 54, large)-net over F25, using
- 255 times duplication [i] based on digital (46, 49, large)-net over F25, using
- t-expansion [i] based on digital (42, 49, large)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (6, 9, large)-net over F25, using
- net defined by OOA [i] based on linear OOA(259, large, F25, 3, 3), using
- appending kth column [i] based on linear OOA(259, large, F25, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(259, large, F25, 3) (dual of [large, large−9, 4]-code), using
- appending kth column [i] based on linear OOA(259, large, F25, 2, 3), using
- net defined by OOA [i] based on linear OOA(259, large, F25, 3, 3), using
- digital (33, 40, 5592400)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (6, 9, large)-net over F25 (see above)
- 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
- (u, u+v)-construction [i] based on
- digital (6, 9, large)-net over F25, using
- (u, u+v)-construction [i] based on
- t-expansion [i] based on digital (42, 49, large)-net over F25, using
(51, 51+3, large)-Net in Base 25 — Upper bound on s
There is no (51, 54, large)-net in base 25, because
- 1 times m-reduction [i] would yield (51, 53, large)-net in base 25, but