Best Known (43−8, 43, s)-Nets in Base 81
(43−8, 43, 4194300)-Net over F81 — Constructive and digital
Digital (35, 43, 4194300)-net over F81, using
- 811 times duplication [i] based on digital (34, 42, 4194300)-net over F81, using
- net defined by OOA [i] based on linear OOA(8142, 4194300, F81, 10, 8) (dual of [(4194300, 10), 41942958, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8142, 8388601, F81, 2, 8) (dual of [(8388601, 2), 16777160, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8142, 8388602, F81, 2, 8) (dual of [(8388602, 2), 16777162, 9]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(8113, large, F81, 4) (dual of [large, large−13, 5]-code), using
- OOA 2-folding [i] based on linear OA(8113, 8388602, F81, 4) (dual of [8388602, 8388589, 5]-code), using
- 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
- linear OOA(8113, 4194301, F81, 2, 4) (dual of [(4194301, 2), 8388589, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8142, 8388602, F81, 2, 8) (dual of [(8388602, 2), 16777162, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(8142, 8388601, F81, 2, 8) (dual of [(8388601, 2), 16777160, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8142, 4194300, F81, 10, 8) (dual of [(4194300, 10), 41942958, 9]-NRT-code), using
(43−8, 43, large)-Net over F81 — Digital
Digital (35, 43, large)-net over F81, using
- t-expansion [i] based on digital (33, 43, large)-net over F81, using
- 2 times m-reduction [i] based on digital (33, 45, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, 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(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- 2 times m-reduction [i] based on digital (33, 45, large)-net over F81, using
(43−8, 43, large)-Net in Base 81 — Upper bound on s
There is no (35, 43, large)-net in base 81, because
- 6 times m-reduction [i] would yield (35, 37, large)-net in base 81, but