MinT - Best Known (175−57, 175, s)-Nets in Base 3

Best Known (175−57, 175, s)-Nets in Base 3

(175−57, 175, 156)-Net over F₃ — Constructive and digital

Digital (118, 175, 156)-net over F₃, using

17 times m-reduction [i] based on digital (118, 192, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 96, 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]

(175−57, 175, 306)-Net over F₃ — Digital

Digital (118, 175, 306)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(175−57, 175, 5183)-Net in Base 3 — Upper bound on s

There is no (118, 175, 5184)-net in base 3, because

1 times m-reduction [i] would yield (118, 174, 5184)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 104595 912689 755077 324437 616509 175065 303488 450154 608080 828818 966828 214857 642564 335105 > 3¹⁷⁴ [i]