MinT - Best Known (9, 9+13, s)-Nets in Base 81

Best Known (9, 9+13, s)-Nets in Base 81

(9, 9+13, 216)-Net over F₈₁ — Constructive and digital

Digital (9, 22, 216)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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]
2. digital (2, 15, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(9, 9+13, 244)-Net over F₈₁ — Digital

Digital (9, 22, 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+13, 178987)-Net in Base 81 — Upper bound on s

There is no (9, 22, 178988)-net in base 81, because

1 times m-reduction [i] would yield (9, 21, 178988)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11972 598109 702689 869691 697917 031813 850241 > 81²¹ [i]