MinT - Best Known (5, 5+9, s)-Nets in Base 81

Best Known (5, 5+9, s)-Nets in Base 81

(5, 5+9, 182)-Net over F₈₁ — Constructive and digital

Digital (5, 14, 182)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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, 10, 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]

(5, 5+9, 44108)-Net in Base 81 — Upper bound on s

There is no (5, 14, 44109)-net in base 81, because

1 times m-reduction [i] would yield (5, 13, 44109)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 6 461280 528949 380447 842881 > 81¹³ [i]