MinT - Best Known (158, 158+78, s)-Nets in Base 3

Best Known (158, 158+78, s)-Nets in Base 3

(158, 158+78, 164)-Net over F₃ — Constructive and digital

Digital (158, 236, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 46, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (112, 190, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 95, 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]

(158, 158+78, 377)-Net over F₃ — Digital

Digital (158, 236, 377)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(158, 158+78, 5898)-Net in Base 3 — Upper bound on s

There is no (158, 236, 5899)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 39886 846063 035399 530621 329589 750447 718385 973953 049639 701395 739704 272541 096676 556936 039472 934012 780090 859085 226339 > 3²³⁶ [i]