Best Known (66, 66+8, s)-Nets in Base 7
(66, 66+8, 4194301)-Net over F7 — Constructive and digital
Digital (66, 74, 4194301)-net over F7, using
- net defined by OOA [i] based on linear OOA(774, 4194301, F7, 10, 8) (dual of [(4194301, 10), 41942936, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(774, large, F7, 2, 8), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(772, 8388602, F7, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- trace code [i] based on linear OOA(4936, 4194301, F49, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4936, 8388602, F49, 8) (dual of [8388602, 8388566, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4936, large, F49, 8) (dual of [large, large−36, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 25679568 | 495−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(4936, large, F49, 8) (dual of [large, large−36, 9]-code), using
- OOA 2-folding [i] based on linear OA(4936, 8388602, F49, 8) (dual of [8388602, 8388566, 9]-code), using
- trace code [i] based on linear OOA(4936, 4194301, F49, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(772, 8388602, F7, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(774, large, F7, 2, 8), using
(66, 66+8, large)-Net over F7 — Digital
Digital (66, 74, large)-net over F7, using
- 71 times duplication [i] based on digital (65, 73, large)-net over F7, using
- t-expansion [i] based on digital (63, 73, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(773, large, F7, 10) (dual of [large, large−73, 11]-code), using
- t-expansion [i] based on digital (63, 73, large)-net over F7, using
(66, 66+8, large)-Net in Base 7 — Upper bound on s
There is no (66, 74, large)-net in base 7, because
- 6 times m-reduction [i] would yield (66, 68, large)-net in base 7, but