Best Known (56, 64, s)-Nets in Base 9
(56, 64, 4194336)-Net over F9 — Constructive and digital
Digital (56, 64, 4194336)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 36)-net over F9, using
- net defined by OOA [i] based on linear OOA(96, 36, F9, 4, 4) (dual of [(36, 4), 138, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(96, 36, F9, 3, 4) (dual of [(36, 3), 102, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- appending kth column [i] based on linear OOA(96, 36, F9, 3, 4) (dual of [(36, 3), 102, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(96, 36, F9, 4, 4) (dual of [(36, 4), 138, 5]-NRT-code), 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
- digital (2, 6, 36)-net over F9, using
(56, 64, large)-Net over F9 — Digital
Digital (56, 64, large)-net over F9, using
- t-expansion [i] based on digital (55, 64, large)-net over F9, using
- 1 times m-reduction [i] based on digital (55, 65, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, 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(965, large, F9, 10) (dual of [large, large−65, 11]-code), using
- 1 times m-reduction [i] based on digital (55, 65, large)-net over F9, using
(56, 64, large)-Net in Base 9 — Upper bound on s
There is no (56, 64, large)-net in base 9, because
- 6 times m-reduction [i] would yield (56, 58, large)-net in base 9, but