MinT - Best Known (90, 90+41, s)-Nets in Base 3

Best Known (90, 90+41, s)-Nets in Base 3

(90, 90+41, 156)-Net over F₃ — Constructive and digital

Digital (90, 131, 156)-net over F₃, using

5 times m-reduction [i] based on digital (90, 136, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 68, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(90, 90+41, 271)-Net over F₃ — Digital

Digital (90, 131, 271)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(90, 90+41, 5223)-Net in Base 3 — Upper bound on s

There is no (90, 131, 5224)-net in base 3, because

1 times m-reduction [i] would yield (90, 130, 5224)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 106 377694 885580 914107 581790 871186 458838 718699 438233 349743 545921 > 3¹³⁰ [i]