Best Known (37, 48, s)-Nets in Base 81
(37, 48, 1680960)-Net over F81 — Constructive and digital
Digital (37, 48, 1680960)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 3240)-net over F81, using
- net defined by OOA [i] based on linear OOA(817, 3240, F81, 5, 5) (dual of [(3240, 5), 16193, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(817, 3240, F81, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- appending kth column [i] based on linear OOA(817, 3240, F81, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(817, 3240, F81, 5, 5) (dual of [(3240, 5), 16193, 6]-NRT-code), using
- digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (2, 7, 3240)-net over F81, using
(37, 48, large)-Net over F81 — Digital
Digital (37, 48, large)-net over F81, using
- t-expansion [i] based on digital (36, 48, large)-net over F81, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (36, 49, large)-net over F81, using
(37, 48, large)-Net in Base 81 — Upper bound on s
There is no (37, 48, large)-net in base 81, because
- 9 times m-reduction [i] would yield (37, 39, large)-net in base 81, but