MinT - Best Known (31−20, 31, s)-Nets in Base 81

Best Known (31−20, 31, s)-Nets in Base 81

(31−20, 31, 182)-Net over F₈₁ — Constructive and digital

Digital (11, 31, 182)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 10, 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, 21, 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]

(31−20, 31, 244)-Net over F₈₁ — Digital

Digital (11, 31, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 31, 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]

(31−20, 31, 46681)-Net in Base 81 — Upper bound on s

There is no (11, 31, 46682)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 145566 561903 076047 246186 076097 839031 778676 131487 463141 569601 > 81³¹ [i]