Best Known (46, 46+13, s)-Nets in Base 81
(46, 46+13, 1398282)-Net over F81 — Constructive and digital
Digital (46, 59, 1398282)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 182)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (1, 7, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (0, 3, 82)-net over F81, using
- (u, u+v)-construction [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
- digital (4, 10, 182)-net over F81, using
(46, 46+13, large)-Net over F81 — Digital
Digital (46, 59, large)-net over F81, using
- t-expansion [i] based on digital (45, 59, large)-net over F81, using
- 2 times m-reduction [i] based on digital (45, 61, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- 2 times m-reduction [i] based on digital (45, 61, large)-net over F81, using
(46, 46+13, large)-Net in Base 81 — Upper bound on s
There is no (46, 59, large)-net in base 81, because
- 11 times m-reduction [i] would yield (46, 48, large)-net in base 81, but