Best Known (77, 77+8, s)-Nets in Base 9
(77, 77+8, large)-Net over F9 — Constructive and digital
Digital (77, 85, large)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 10)-net over F9, using
- digital (21, 25, 4194300)-net over F9, using
- s-reduction based on digital (21, 25, 4194301)-net over F9, using
- net defined by OOA [i] based on linear OOA(925, 4194301, F9, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(925, 4194301, F9, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(925, 8388602, F9, 4) (dual of [8388602, 8388577, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(925, large, F9, 4) (dual of [large, large−25, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(925, large, F9, 4) (dual of [large, large−25, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(925, 8388602, F9, 4) (dual of [8388602, 8388577, 5]-code), using
- appending kth column [i] based on linear OOA(925, 4194301, F9, 3, 4) (dual of [(4194301, 3), 12582878, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(925, 4194301, F9, 4, 4) (dual of [(4194301, 4), 16777179, 5]-NRT-code), using
- s-reduction based on digital (21, 25, 4194301)-net over F9, using
- digital (50, 58, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
(77, 77+8, large)-Net in Base 9 — Upper bound on s
There is no (77, 85, large)-net in base 9, because
- 6 times m-reduction [i] would yield (77, 79, large)-net in base 9, but