Best Known (27−7, 27, s)-Nets in Base 81
(27−7, 27, 2796200)-Net over F81 — Constructive and digital
Digital (20, 27, 2796200)-net over F81, using
- 812 times duplication [i] based on digital (18, 25, 2796200)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8125, 8388601, F81, 7) (dual of [8388601, 8388576, 8]-code), using
- net defined by OOA [i] based on linear OOA(8125, 2796200, F81, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
(27−7, 27, large)-Net over F81 — Digital
Digital (20, 27, large)-net over F81, using
- 812 times duplication [i] based on digital (18, 25, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
(27−7, 27, large)-Net in Base 81 — Upper bound on s
There is no (20, 27, large)-net in base 81, because
- 5 times m-reduction [i] would yield (20, 22, large)-net in base 81, but