Best Known (37, 37+13, s)-Nets in Base 81
(37, 37+13, 1398100)-Net over F81 — Constructive and digital
Digital (37, 50, 1398100)-net over F81, using
- 811 times duplication [i] based on digital (36, 49, 1398100)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8149, 8388601, F81, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(8149, 1398100, F81, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
(37, 37+13, large)-Net over F81 — Digital
Digital (37, 50, large)-net over F81, using
- 811 times duplication [i] based on digital (36, 49, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
(37, 37+13, large)-Net in Base 81 — Upper bound on s
There is no (37, 50, large)-net in base 81, because
- 11 times m-reduction [i] would yield (37, 39, large)-net in base 81, but