MinT - Best Known (239−77, 239, s)-Nets in Base 3

Best Known (239−77, 239, s)-Nets in Base 3

(239−77, 239, 172)-Net over F₃ — Constructive and digital

Digital (162, 239, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 51, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (111, 188, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 94, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(239−77, 239, 413)-Net over F₃ — Digital

Digital (162, 239, 413)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−77, 239, 7274)-Net in Base 3 — Upper bound on s

There is no (162, 239, 7275)-net in base 3, because

1 times m-reduction [i] would yield (162, 238, 7275)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 358874 019189 637313 430174 991113 584050 618346 594907 225478 280972 292581 319867 215032 743501 873466 748843 757645 385440 579053 > 3²³⁸ [i]