Best Known (67, 67+7, s)-Nets in Base 5
(67, 67+7, 5713140)-Net over F5 — Constructive and digital
Digital (67, 74, 5713140)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (9, 12, 120740)-net over F5, using
- net defined by OOA [i] based on linear OOA(512, 120740, F5, 3, 3) (dual of [(120740, 3), 362208, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(512, 120740, F5, 2, 3) (dual of [(120740, 2), 241468, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(512, 120740, F5, 3) (dual of [120740, 120728, 4]-code or 120740-cap in PG(11,5)), using
- appending kth column [i] based on linear OOA(512, 120740, F5, 2, 3) (dual of [(120740, 2), 241468, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(512, 120740, F5, 3, 3) (dual of [(120740, 3), 362208, 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 (9, 12, 120740)-net over F5, using
(67, 67+7, large)-Net over F5 — Digital
Digital (67, 74, large)-net over F5, using
- 52 times duplication [i] based on digital (65, 72, large)-net over F5, using
- t-expansion [i] based on digital (63, 72, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times code embedding in larger space [i] based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
- t-expansion [i] based on digital (63, 72, large)-net over F5, using
(67, 67+7, large)-Net in Base 5 — Upper bound on s
There is no (67, 74, large)-net in base 5, because
- 5 times m-reduction [i] would yield (67, 69, large)-net in base 5, but