Best Known (54−22, 54, s)-Nets in Base 81
(54−22, 54, 678)-Net over F81 — Constructive and digital
Digital (32, 54, 678)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (21, 43, 596)-net over F81, using
- net defined by OOA [i] based on linear OOA(8143, 596, F81, 22, 22) (dual of [(596, 22), 13069, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8143, 6556, F81, 22) (dual of [6556, 6513, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8143, 6556, F81, 22) (dual of [6556, 6513, 23]-code), using
- net defined by OOA [i] based on linear OOA(8143, 596, F81, 22, 22) (dual of [(596, 22), 13069, 23]-NRT-code), using
- digital (0, 11, 82)-net over F81, using
(54−22, 54, 8779)-Net over F81 — Digital
Digital (32, 54, 8779)-net over F81, using
(54−22, 54, large)-Net in Base 81 — Upper bound on s
There is no (32, 54, large)-net in base 81, because
- 20 times m-reduction [i] would yield (32, 34, large)-net in base 81, but