Best Known (41, 41+9, s)-Nets in Base 25
(41, 41+9, 2098126)-Net over F25 — Constructive and digital
Digital (41, 50, 2098126)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (5, 9, 976)-net over F25, using
- net defined by OOA [i] based on linear OOA(259, 976, F25, 4, 4) (dual of [(976, 4), 3895, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(259, 1952, F25, 4) (dual of [1952, 1943, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(259, 1953, F25, 4) (dual of [1953, 1944, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(259, 1952, F25, 4) (dual of [1952, 1943, 5]-code), using
- net defined by OOA [i] based on linear OOA(259, 976, F25, 4, 4) (dual of [(976, 4), 3895, 5]-NRT-code), 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
- digital (5, 9, 976)-net over F25, using
(41, 41+9, large)-Net over F25 — Digital
Digital (41, 50, large)-net over F25, using
- t-expansion [i] based on digital (40, 50, large)-net over F25, using
- 1 times m-reduction [i] based on digital (40, 51, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- 1 times m-reduction [i] based on digital (40, 51, large)-net over F25, using
(41, 41+9, large)-Net in Base 25 — Upper bound on s
There is no (41, 50, large)-net in base 25, because
- 7 times m-reduction [i] would yield (41, 43, large)-net in base 25, but