Best Known (66−16, 66, s)-Nets in Base 81
(66−16, 66, 1048575)-Net over F81 — Constructive and digital
Digital (50, 66, 1048575)-net over F81, using
- 811 times duplication [i] based on digital (49, 65, 1048575)-net over F81, using
- t-expansion [i] based on digital (48, 65, 1048575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- t-expansion [i] based on digital (48, 65, 1048575)-net over F81, using
(66−16, 66, large)-Net over F81 — Digital
Digital (50, 66, large)-net over F81, using
- 811 times duplication [i] based on digital (49, 65, large)-net over F81, using
- t-expansion [i] based on digital (48, 65, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- t-expansion [i] based on digital (48, 65, large)-net over F81, using
(66−16, 66, large)-Net in Base 81 — Upper bound on s
There is no (50, 66, large)-net in base 81, because
- 14 times m-reduction [i] would yield (50, 52, large)-net in base 81, but