MinT - Best Known (9, 9+47, s)-Nets in Base 81

Best Known (9, 9+47, s)-Nets in Base 81

(9, 9+47, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 56, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 56, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(9, 9+47, 244)-Net over F₈₁ — Digital

Digital (9, 56, 244)-net over F₈₁, using

net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
  - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(9, 9+47, 4305)-Net in Base 81 — Upper bound on s

There is no (9, 56, 4306)-net in base 81, because

1 times m-reduction [i] would yield (9, 55, 4306)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 929 852558 150558 579864 430042 511115 111528 077830 745967 217294 191390 143804 560430 865877 663455 800512 267193 478241 > 81⁵⁵ [i]