Best Known (31−8, 31, s)-Nets in Base 81
(31−8, 31, 2097150)-Net over F81 — Constructive and digital
Digital (23, 31, 2097150)-net over F81, using
- 812 times duplication [i] based on digital (21, 29, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 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
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
(31−8, 31, large)-Net over F81 — Digital
Digital (23, 31, large)-net over F81, using
- 812 times duplication [i] based on digital (21, 29, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
(31−8, 31, large)-Net in Base 81 — Upper bound on s
There is no (23, 31, large)-net in base 81, because
- 6 times m-reduction [i] would yield (23, 25, large)-net in base 81, but