MinT - Best Known (10, 10+19, s)-Nets in Base 81

Best Known (10, 10+19, s)-Nets in Base 81

(10, 10+19, 182)-Net over F₈₁ — Constructive and digital

Digital (10, 29, 182)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 9, 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, 20, 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]

(10, 10+19, 244)-Net over F₈₁ — Digital

Digital (10, 29, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 29, 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]

(10, 10+19, 44888)-Net in Base 81 — Upper bound on s

There is no (10, 29, 44889)-net in base 81, because

1 times m-reduction [i] would yield (10, 28, 44889)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 273945 576874 393767 938750 271997 962883 439545 256614 592081 > 81²⁸ [i]