MinT - Best Known (163, 163+77, s)-Nets in Base 3

Best Known (163, 163+77, s)-Nets in Base 3

(163, 163+77, 172)-Net over F₃ — Constructive and digital

Digital (163, 240, 172)-net over F₃, using

3¹ times duplication [i] based on 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]

(163, 163+77, 420)-Net over F₃ — Digital

Digital (163, 240, 420)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+77, 7489)-Net in Base 3 — Upper bound on s

There is no (163, 240, 7490)-net in base 3, because

1 times m-reduction [i] would yield (163, 239, 7490)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 079388 572528 603163 375698 739166 238432 222577 854681 469303 590705 392326 174134 818968 470619 924810 019376 172227 715843 859197 > 3²³⁹ [i]