MinT - Best Known (12, 31, s)-Nets in Base 81

Best Known (12, 31, s)-Nets in Base 81

(12, 31, 216)-Net over F₈₁ — Constructive and digital

Digital (12, 31, 216)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (2, 21, 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]

(12, 31, 298)-Net over F₈₁ — Digital

Digital (12, 31, 298)-net over F₈₁, using

net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
  - a curve from the manYPoints database [i]

(12, 31, 119196)-Net in Base 81 — Upper bound on s

There is no (12, 31, 119197)-net in base 81, because

1 times m-reduction [i] would yield (12, 30, 119197)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1797 089170 669939 838894 352235 577547 723629 856999 587217 905041 > 81³⁰ [i]