MinT - Best Known (157, 157+77, s)-Nets in Base 3

Best Known (157, 157+77, s)-Nets in Base 3

(157, 157+77, 164)-Net over F₃ — Constructive and digital

Digital (157, 234, 164)-net over F₃, using

3¹ times duplication [i] based on digital (156, 233, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 45, 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 (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]

(157, 157+77, 379)-Net over F₃ — Digital

Digital (157, 234, 379)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+77, 6290)-Net in Base 3 — Upper bound on s

There is no (157, 234, 6291)-net in base 3, because

1 times m-reduction [i] would yield (157, 233, 6291)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1478 454457 226636 577586 520241 705475 479765 465774 843587 336545 058996 906159 628804 546159 895568 269327 615137 119439 123965 > 3²³³ [i]