Best Known (66−7, 66, s)-Nets in Base 5
(66−7, 66, 5592426)-Net over F5 — Constructive and digital
Digital (59, 66, 5592426)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 26)-net over F5, using
- net defined by OOA [i] based on linear OOA(54, 26, F5, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
- digital (55, 62, 5592400)-net over F5, using
- trace code for nets [i] based on 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
- trace code for nets [i] based on digital (24, 31, 2796200)-net over F25, using
- digital (1, 4, 26)-net over F5, using
(66−7, 66, large)-Net over F5 — Digital
Digital (59, 66, large)-net over F5, using
- 54 times duplication [i] based on digital (55, 62, large)-net over F5, using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- 1 times code embedding in larger space [i] based on linear OA(561, large, F5, 8) (dual of [large, large−61, 9]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(562, large, F5, 8) (dual of [large, large−62, 9]-code), using
- t-expansion [i] based on digital (54, 62, large)-net over F5, using
(66−7, 66, large)-Net in Base 5 — Upper bound on s
There is no (59, 66, large)-net in base 5, because
- 5 times m-reduction [i] would yield (59, 61, large)-net in base 5, but