MinT - Best Known (9, 9+17, s)-Nets in Base 81

Best Known (9, 9+17, s)-Nets in Base 81

(9, 9+17, 182)-Net over F₈₁ — Constructive and digital

Digital (9, 26, 182)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 8, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ 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]
2. digital (1, 18, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]

(9, 9+17, 244)-Net over F₈₁ — Digital

Digital (9, 26, 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+17, 43309)-Net in Base 81 — Upper bound on s

There is no (9, 26, 43310)-net in base 81, because

1 times m-reduction [i] would yield (9, 25, 43310)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 515458 397340 615202 702226 635135 506418 274135 494401 > 81²⁵ [i]