Best Known (85−11, 85, s)-Nets in Base 25
(85−11, 85, 5049436)-Net over F25 — Constructive and digital
Digital (74, 85, 5049436)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 4, 16276)-net over F25, using
- digital (6, 9, 1677720)-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 (16, 21, 1677720)-net over F25, using
- s-reduction based on digital (16, 21, 4194301)-net over F25, using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(2521, large, F25, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(2521, 4194301, F25, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- s-reduction based on digital (16, 21, 4194301)-net over F25, using
- digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
(85−11, 85, large)-Net over F25 — Digital
Digital (74, 85, large)-net over F25, using
- t-expansion [i] based on digital (73, 85, large)-net over F25, using
- 7 times m-reduction [i] based on digital (73, 92, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2592, large, F25, 19) (dual of [large, large−92, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2591, large, F25, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(2591, large, F25, 19) (dual of [large, large−91, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2592, large, F25, 19) (dual of [large, large−92, 20]-code), using
- 7 times m-reduction [i] based on digital (73, 92, large)-net over F25, using
(85−11, 85, large)-Net in Base 25 — Upper bound on s
There is no (74, 85, large)-net in base 25, because
- 9 times m-reduction [i] would yield (74, 76, large)-net in base 25, but