Best Known (42, 53, s)-Nets in Base 81
(42, 53, 1690681)-Net over F81 — Constructive and digital
Digital (42, 53, 1690681)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (7, 12, 12961)-net over F81, using
- net defined by OOA [i] based on linear OOA(8112, 12961, F81, 5, 5) (dual of [(12961, 5), 64793, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8112, 12961, F81, 4, 5) (dual of [(12961, 4), 51832, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8112, 25923, F81, 5) (dual of [25923, 25911, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(8112, 25924, F81, 5) (dual of [25924, 25912, 6]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(811, 6481, F81, 1) (dual of [6481, 6480, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(811, 6481, F81, 1) (dual of [6481, 6480, 2]-code) (see above)
- linear OA(813, 6481, F81, 2) (dual of [6481, 6478, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(813, 6643, F81, 2) (dual of [6643, 6640, 3]-code), using
- Hamming code H(3,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 6643, F81, 2) (dual of [6643, 6640, 3]-code), using
- linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- linear OA(811, 6481, F81, 1) (dual of [6481, 6480, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(8112, 25924, F81, 5) (dual of [25924, 25912, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8112, 25923, F81, 5) (dual of [25923, 25911, 6]-code), using
- appending kth column [i] based on linear OOA(8112, 12961, F81, 4, 5) (dual of [(12961, 4), 51832, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8112, 12961, F81, 5, 5) (dual of [(12961, 5), 64793, 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 (7, 12, 12961)-net over F81, using
(42, 53, large)-Net over F81 — Digital
Digital (42, 53, large)-net over F81, using
- 4 times m-reduction [i] based on digital (42, 57, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
(42, 53, large)-Net in Base 81 — Upper bound on s
There is no (42, 53, large)-net in base 81, because
- 9 times m-reduction [i] would yield (42, 44, large)-net in base 81, but