Best Known (52−11, 52, s)-Nets in Base 81
(52−11, 52, 1687441)-Net over F81 — Constructive and digital
Digital (41, 52, 1687441)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (6, 11, 9721)-net over F81, using
- net defined by OOA [i] based on linear OOA(8111, 9721, F81, 5, 5) (dual of [(9721, 5), 48594, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(8111, 9721, F81, 4, 5) (dual of [(9721, 4), 38873, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(8111, 19443, F81, 5) (dual of [19443, 19432, 6]-code), using
- (u, u−v, u+v+w)-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(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
- (u, u−v, u+v+w)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(8111, 19443, F81, 5) (dual of [19443, 19432, 6]-code), using
- appending kth column [i] based on linear OOA(8111, 9721, F81, 4, 5) (dual of [(9721, 4), 38873, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8111, 9721, F81, 5, 5) (dual of [(9721, 5), 48594, 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 (6, 11, 9721)-net over F81, using
(52−11, 52, large)-Net over F81 — Digital
Digital (41, 52, large)-net over F81, using
- t-expansion [i] based on digital (39, 52, large)-net over F81, using
- 1 times m-reduction [i] based on digital (39, 53, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F81, 14) (dual of [large, large−53, 15]-code), using
- 1 times m-reduction [i] based on digital (39, 53, large)-net over F81, using
(52−11, 52, large)-Net in Base 81 — Upper bound on s
There is no (41, 52, large)-net in base 81, because
- 9 times m-reduction [i] would yield (41, 43, large)-net in base 81, but