Best Known (67, 75, s)-Nets in Base 5
(67, 75, 4194301)-Net over F5 — Constructive and digital
Digital (67, 75, 4194301)-net over F5, using
- 51 times duplication [i] based on digital (66, 74, 4194301)-net over F5, using
- net defined by OOA [i] based on linear OOA(574, 4194301, F5, 10, 8) (dual of [(4194301, 10), 41942936, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(574, large, F5, 2, 8), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2536, large, F25, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(2536, 8388602, F25, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(2536, 4194301, F25, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(572, 8388602, F5, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(574, large, F5, 2, 8), using
- net defined by OOA [i] based on linear OOA(574, 4194301, F5, 10, 8) (dual of [(4194301, 10), 41942936, 9]-NRT-code), using
(67, 75, large)-Net over F5 — Digital
Digital (67, 75, large)-net over F5, using
- 53 times duplication [i] based on digital (64, 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, 75, large)-Net in Base 5 — Upper bound on s
There is no (67, 75, large)-net in base 5, because
- 6 times m-reduction [i] would yield (67, 69, large)-net in base 5, but