Best Known (65, 81, s)-Nets in Base 25
(65, 81, 1048575)-Net over F25 — Constructive and digital
Digital (65, 81, 1048575)-net over F25, using
- t-expansion [i] based on digital (64, 81, 1048575)-net over F25, using
- net defined by OOA [i] based on linear OOA(2581, 1048575, F25, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2581, 8388601, F25, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2581, 8388601, F25, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2581, 1048575, F25, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
(65, 81, large)-Net over F25 — Digital
Digital (65, 81, large)-net over F25, using
- 1 times m-reduction [i] based on digital (65, 82, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2582, large, F25, 17) (dual of [large, large−82, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(2581, large, F25, 17) (dual of [large, large−81, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2582, large, F25, 17) (dual of [large, large−82, 18]-code), using
(65, 81, large)-Net in Base 25 — Upper bound on s
There is no (65, 81, large)-net in base 25, because
- 14 times m-reduction [i] would yield (65, 67, large)-net in base 25, but