MinT - Best Known (27, 68, s)-Nets in Base 81

Best Known (27, 68, s)-Nets in Base 81

(27, 68, 370)-Net over F₈₁ — Constructive and digital

Digital (27, 68, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 68, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(27, 68, 500)-Net over F₈₁ — Digital

Digital (27, 68, 500)-net over F₈₁, using

t-expansion [i] based on digital (26, 68, 500)-net over F₈₁, using
- net from sequence [i] based on digital (26, 499)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 26 and N(F) ≥ 500, using
    - a curve from the manYPoints database [i]

(27, 68, 256817)-Net in Base 81 — Upper bound on s

There is no (27, 68, 256818)-net in base 81, because

1 times m-reduction [i] would yield (27, 67, 256818)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 73 877818 246667 809464 819262 323307 364423 881738 314976 667503 165484 815069 959669 511920 506408 079844 988812 304903 344185 396421 959863 644801 > 81⁶⁷ [i]