MinT - Best Known (231−45, 231, s)-Nets in Base 3

Best Known (231−45, 231, s)-Nets in Base 3

(231−45, 231, 695)-Net over F₃ — Constructive and digital

Digital (186, 231, 695)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (163, 208, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 52, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(231−45, 231, 2781)-Net over F₃ — Digital

Digital (186, 231, 2781)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−45, 231, 440440)-Net in Base 3 — Upper bound on s

There is no (186, 231, 440441)-net in base 3, because

1 times m-reduction [i] would yield (186, 230, 440441)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 687739 285941 545361 834632 053193 398586 346731 808801 982802 261427 588923 472225 481512 342861 041432 884872 195895 282377 > 3²³⁰ [i]