Best Known (68−9, 68, s)-Nets in Base 25
(68−9, 68, 6291476)-Net over F25 — Constructive and digital
Digital (59, 68, 6291476)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 26)-net over F25, using
- digital (6, 9, 2097150)-net over F25, using
- s-reduction 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
- s-reduction based on digital (6, 9, large)-net over F25, using
- digital (12, 16, 2097150)-net over F25, using
- s-reduction based on digital (12, 16, 4194301)-net over F25, using
- net defined by OOA [i] based on linear OOA(2516, 4194301, F25, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2516, 8388602, F25, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2516, large, F25, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2516, large, F25, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2516, 8388602, F25, 4) (dual of [8388602, 8388586, 5]-code), using
- net defined by OOA [i] based on linear OOA(2516, 4194301, F25, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- s-reduction based on digital (12, 16, 4194301)-net over F25, using
- digital (32, 41, 2097150)-net over F25, using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2541, large, F25, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2541, 8388601, F25, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2541, 2097150, F25, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
(68−9, 68, large)-Net over F25 — Digital
Digital (59, 68, large)-net over F25, using
- t-expansion [i] based on digital (57, 68, large)-net over F25, using
- 4 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
- 4 times m-reduction [i] based on digital (57, 72, large)-net over F25, using
(68−9, 68, large)-Net in Base 25 — Upper bound on s
There is no (59, 68, large)-net in base 25, because
- 7 times m-reduction [i] would yield (59, 61, large)-net in base 25, but