MinT - Best Known (231−109, 231, s)-Nets in Base 3

Best Known (231−109, 231, s)-Nets in Base 3

(231−109, 231, 80)-Net over F₃ — Constructive and digital

Digital (122, 231, 80)-net over F₃, using

3 times m-reduction [i] based on digital (122, 234, 80)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 77, 32)-net over F₃, using
    - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 157, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(231−109, 231, 134)-Net over F₃ — Digital

Digital (122, 231, 134)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−109, 231, 1076)-Net in Base 3 — Upper bound on s

There is no (122, 231, 1077)-net in base 3, because

1 times m-reduction [i] would yield (122, 230, 1077)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55 916198 265298 552761 922307 378677 075031 889000 291866 548429 243753 704834 319339 657927 752871 466062 700715 969305 973729 > 3²³⁰ [i]