MinT - Best Known (137, 137+86, s)-Nets in Base 3

Best Known (137, 137+86, s)-Nets in Base 3

(137, 137+86, 156)-Net over F₃ — Constructive and digital

Digital (137, 223, 156)-net over F₃, using

7 times m-reduction [i] based on digital (137, 230, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 115, 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]

(137, 137+86, 228)-Net over F₃ — Digital

Digital (137, 223, 228)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+86, 2474)-Net in Base 3 — Upper bound on s

There is no (137, 223, 2475)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25271 396330 131808 243271 960441 281865 014113 830088 382940 395939 780064 903036 040882 869625 372072 254210 951475 722619 > 3²²³ [i]