Best Known (34−9, 34, s)-Nets in Base 81
(34−9, 34, 2097150)-Net over F81 — Constructive and digital
Digital (25, 34, 2097150)-net over F81, using
- 811 times duplication [i] based on digital (24, 33, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 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 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
(34−9, 34, large)-Net over F81 — Digital
Digital (25, 34, large)-net over F81, using
- 811 times duplication [i] based on digital (24, 33, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
(34−9, 34, large)-Net in Base 81 — Upper bound on s
There is no (25, 34, large)-net in base 81, because
- 7 times m-reduction [i] would yield (25, 27, large)-net in base 81, but