MinT - Best Known (117, 117+55, s)-Nets in Base 3

Best Known (117, 117+55, s)-Nets in Base 3

(117, 117+55, 156)-Net over F₃ — Constructive and digital

Digital (117, 172, 156)-net over F₃, using

18 times m-reduction [i] based on digital (117, 190, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 95, 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]

(117, 117+55, 319)-Net over F₃ — Digital

Digital (117, 172, 319)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(117, 117+55, 5716)-Net in Base 3 — Upper bound on s

There is no (117, 172, 5717)-net in base 3, because

1 times m-reduction [i] would yield (117, 171, 5717)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3876 965074 479239 302614 732226 332741 788227 124227 589866 328242 259553 769890 269934 874379 > 3¹⁷¹ [i]