Best Known (51−14, 51, s)-Nets in Base 81
(51−14, 51, 76102)-Net over F81 — Constructive and digital
Digital (37, 51, 76102)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 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, 8, 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 (26, 40, 75920)-net over F81, using
- net defined by OOA [i] based on linear OOA(8140, 75920, F81, 14, 14) (dual of [(75920, 14), 1062840, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(8140, 531440, F81, 14) (dual of [531440, 531400, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(8140, 531440, F81, 14) (dual of [531440, 531400, 15]-code), using
- net defined by OOA [i] based on linear OOA(8140, 75920, F81, 14, 14) (dual of [(75920, 14), 1062840, 15]-NRT-code), using
- digital (4, 11, 182)-net over F81, using
(51−14, 51, 2174959)-Net over F81 — Digital
Digital (37, 51, 2174959)-net over F81, using
(51−14, 51, large)-Net in Base 81 — Upper bound on s
There is no (37, 51, large)-net in base 81, because
- 12 times m-reduction [i] would yield (37, 39, large)-net in base 81, but