MinT - Best Known (159, 159+75, s)-Nets in Base 3

Best Known (159, 159+75, s)-Nets in Base 3

(159, 159+75, 172)-Net over F₃ — Constructive and digital

Digital (159, 234, 172)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 50, 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 (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]

(159, 159+75, 412)-Net over F₃ — Digital

Digital (159, 234, 412)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+75, 7367)-Net in Base 3 — Upper bound on s

There is no (159, 234, 7368)-net in base 3, because

1 times m-reduction [i] would yield (159, 233, 7368)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1480 991639 508504 799633 269469 222837 780927 378552 702948 144975 505496 006569 776974 375943 506374 932701 442545 506104 861969 > 3²³³ [i]