MinT - Best Known (167, 167+39, s)-Nets in Base 3

Best Known (167, 167+39, s)-Nets in Base 3

(167, 167+39, 698)-Net over F₃ — Constructive and digital

Digital (167, 206, 698)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 22, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (145, 184, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 46, 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]

(167, 167+39, 2918)-Net over F₃ — Digital

Digital (167, 206, 2918)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167, 167+39, 557260)-Net in Base 3 — Upper bound on s

There is no (167, 206, 557261)-net in base 3, because

1 times m-reduction [i] would yield (167, 205, 557261)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64 545144 038254 396455 883672 160892 322267 164656 589889 383719 346918 702225 576108 548787 313096 152707 549451 > 3²⁰⁵ [i]