MinT - Best Known (212−81, 212, s)-Nets in Base 3

Best Known (212−81, 212, s)-Nets in Base 3

(212−81, 212, 156)-Net over F₃ — Constructive and digital

Digital (131, 212, 156)-net over F₃, using

6 times m-reduction [i] based on digital (131, 218, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 109, 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]

(212−81, 212, 224)-Net over F₃ — Digital

Digital (131, 212, 224)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−81, 212, 2552)-Net in Base 3 — Upper bound on s

There is no (131, 212, 2553)-net in base 3, because

1 times m-reduction [i] would yield (131, 211, 2553)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47493 637876 981691 305551 582925 730326 413939 838386 052325 686590 570321 541845 140747 467448 235166 884449 305889 > 3²¹¹ [i]