Best Known (69, 78, s)-Nets in Base 9
(69, 78, 4200780)-Net over F9 — Constructive and digital
Digital (69, 78, 4200780)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 6480)-net over F9, using
- net defined by OOA [i] based on linear OOA(912, 6480, F9, 4, 4) (dual of [(6480, 4), 25908, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(912, 6480, F9, 3, 4) (dual of [(6480, 3), 19428, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(912, 12960, F9, 4) (dual of [12960, 12948, 5]-code), using
- trace code [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- 1 times truncation [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- trace code [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(912, 12960, F9, 4) (dual of [12960, 12948, 5]-code), using
- appending kth column [i] based on linear OOA(912, 6480, F9, 3, 4) (dual of [(6480, 3), 19428, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(912, 6480, F9, 4, 4) (dual of [(6480, 4), 25908, 5]-NRT-code), using
- digital (57, 66, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 2-folding [i] based on linear OA(8133, 8388602, F81, 9) (dual of [8388602, 8388569, 10]-code), using
- trace code [i] based on linear OOA(8133, 4194301, F81, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(966, 8388602, F9, 2, 9) (dual of [(8388602, 2), 16777138, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(966, 8388601, F9, 2, 9) (dual of [(8388601, 2), 16777136, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(966, 4194300, F9, 10, 9) (dual of [(4194300, 10), 41942934, 10]-NRT-code), using
- digital (8, 12, 6480)-net over F9, using
(69, 78, large)-Net over F9 — Digital
Digital (69, 78, large)-net over F9, using
- 3 times m-reduction [i] based on digital (69, 81, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
(69, 78, large)-Net in Base 9 — Upper bound on s
There is no (69, 78, large)-net in base 9, because
- 7 times m-reduction [i] would yield (69, 71, large)-net in base 9, but