MinT - Best Known (157, 157+76, s)-Nets in Base 3

Best Known (157, 157+76, s)-Nets in Base 3

(157, 157+76, 167)-Net over F₃ — Constructive and digital

Digital (157, 233, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 47, 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 (110, 186, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 93, 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+76, 389)-Net over F₃ — Digital

Digital (157, 233, 389)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+76, 6290)-Net in Base 3 — Upper bound on s

There is no (157, 233, 6291)-net in base 3, because

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]