Best Known (54, 54+20, s)-Nets in Base 81
(54, 54+20, 53326)-Net over F81 — Constructive and digital
Digital (54, 74, 53326)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 182)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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, 11, 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, 5, 82)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (38, 58, 53144)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 53144, F81, 20, 20) (dual of [(53144, 20), 1062822, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8158, 531440, F81, 20) (dual of [531440, 531382, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8158, 531440, F81, 20) (dual of [531440, 531382, 21]-code), using
- net defined by OOA [i] based on linear OOA(8158, 53144, F81, 20, 20) (dual of [(53144, 20), 1062822, 21]-NRT-code), using
- digital (6, 16, 182)-net over F81, using
(54, 54+20, 2686420)-Net over F81 — Digital
Digital (54, 74, 2686420)-net over F81, using
(54, 54+20, large)-Net in Base 81 — Upper bound on s
There is no (54, 74, large)-net in base 81, because
- 18 times m-reduction [i] would yield (54, 56, large)-net in base 81, but