Best Known (81−18, 81, s)-Nets in Base 81
(81−18, 81, 932183)-Net over F81 — Constructive and digital
Digital (63, 81, 932183)-net over F81, using
- 811 times duplication [i] based on digital (62, 80, 932183)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (51, 69, 932067)-net over F81, using
- net defined by OOA [i] based on linear OOA(8169, 932067, F81, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(8169, large, F81, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(8169, 932067, F81, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- digital (2, 11, 116)-net over F81, using
- (u, u+v)-construction [i] based on
(81−18, 81, large)-Net over F81 — Digital
Digital (63, 81, large)-net over F81, using
- t-expansion [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
(81−18, 81, large)-Net in Base 81 — Upper bound on s
There is no (63, 81, large)-net in base 81, because
- 16 times m-reduction [i] would yield (63, 65, large)-net in base 81, but