MinT - Best Known (212−80, 212, s)-Nets in Base 3

Best Known (212−80, 212, s)-Nets in Base 3

(212−80, 212, 156)-Net over F₃ — Constructive and digital

Digital (132, 212, 156)-net over F₃, using

8 times m-reduction [i] based on digital (132, 220, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 110, 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]

(212−80, 212, 232)-Net over F₃ — Digital

Digital (132, 212, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−80, 212, 2624)-Net in Base 3 — Upper bound on s

There is no (132, 212, 2625)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 142102 753083 435452 223986 034876 381199 738352 429197 584428 339049 695268 774472 599867 124207 731941 010528 073761 > 3²¹² [i]