Best Known (46−10, 46, s)-Nets in Base 81
(46−10, 46, 1681042)-Net over F81 — Constructive and digital
Digital (36, 46, 1681042)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 3322)-net over F81, using
- net defined by OOA [i] based on linear OOA(819, 3322, F81, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(819, 3322, F81, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(812, 82, F81, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;326,81) [i]
- 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
- linear OOA(812, 82, F81, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(819, 3322, F81, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(819, 3322, F81, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
- digital (27, 37, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(8137, 8388600, F81, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(8137, 1677720, F81, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (4, 9, 3322)-net over F81, using
(46−10, 46, large)-Net over F81 — Digital
Digital (36, 46, large)-net over F81, using
- 3 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
(46−10, 46, large)-Net in Base 81 — Upper bound on s
There is no (36, 46, large)-net in base 81, because
- 8 times m-reduction [i] would yield (36, 38, large)-net in base 81, but