Best Known (51−21, 51, s)-Nets in Base 81
(51−21, 51, 738)-Net over F81 — Constructive and digital
Digital (30, 51, 738)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (20, 41, 656)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 656, F81, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- net defined by OOA [i] based on linear OOA(8141, 656, F81, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
- digital (0, 10, 82)-net over F81, using
(51−21, 51, 7646)-Net over F81 — Digital
Digital (30, 51, 7646)-net over F81, using
(51−21, 51, large)-Net in Base 81 — Upper bound on s
There is no (30, 51, large)-net in base 81, because
- 19 times m-reduction [i] would yield (30, 32, large)-net in base 81, but