Best Known (67−12, 67, s)-Nets in Base 81
(67−12, 67, 2796200)-Net over F81 — Constructive and digital
Digital (55, 67, 2796200)-net over F81, using
- 811 times duplication [i] based on digital (54, 66, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8166, 2796200, F81, 14, 12) (dual of [(2796200, 14), 39146734, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8166, 8388601, F81, 2, 12) (dual of [(8388601, 2), 16777136, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8166, 8388602, F81, 2, 12) (dual of [(8388602, 2), 16777138, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(8121, 4194301, F81, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8121, 8388602, F81, 6) (dual of [8388602, 8388581, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- OOA 2-folding [i] based on linear OA(8121, 8388602, F81, 6) (dual of [8388602, 8388581, 7]-code), using
- linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- linear OOA(8121, 4194301, F81, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(8166, 8388602, F81, 2, 12) (dual of [(8388602, 2), 16777138, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8166, 8388601, F81, 2, 12) (dual of [(8388601, 2), 16777136, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8166, 2796200, F81, 14, 12) (dual of [(2796200, 14), 39146734, 13]-NRT-code), using
(67−12, 67, large)-Net over F81 — Digital
Digital (55, 67, large)-net over F81, using
- t-expansion [i] based on digital (54, 67, large)-net over F81, using
- 6 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- 6 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
(67−12, 67, large)-Net in Base 81 — Upper bound on s
There is no (55, 67, large)-net in base 81, because
- 10 times m-reduction [i] would yield (55, 57, large)-net in base 81, but