MinT - Best Known (156, 231, s)-Nets in Base 3

Best Known (156, 231, s)-Nets in Base 3

(156, 231, 167)-Net over F₃ — Constructive and digital

Digital (156, 231, 167)-net over F₃, using

3¹ times duplication [i] based on digital (155, 230, 167)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 46, 19)-net over F₃, using
    - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
        an explicitly constructive algebraic function field [i]
  2. digital (109, 184, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 92, 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]

(156, 231, 391)-Net over F₃ — Digital

Digital (156, 231, 391)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 231, 6736)-Net in Base 3 — Upper bound on s

There is no (156, 231, 6737)-net in base 3, because

1 times m-reduction [i] would yield (156, 230, 6737)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 861118 370889 560520 867956 468760 983852 985877 952586 506487 890451 919919 450743 231591 575328 924715 043895 576308 094227 > 3²³⁰ [i]