MinT - Best Known (139, 139+82, s)-Nets in Base 3

Best Known (139, 139+82, s)-Nets in Base 3

(139, 139+82, 156)-Net over F₃ — Constructive and digital

Digital (139, 221, 156)-net over F₃, using

13 times m-reduction [i] based on digital (139, 234, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 117, 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]

(139, 139+82, 253)-Net over F₃ — Digital

Digital (139, 221, 253)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+82, 2970)-Net in Base 3 — Upper bound on s

There is no (139, 221, 2971)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2796 170792 808970 852478 258749 782615 442903 763821 721526 879497 839140 618057 417163 869280 733030 217056 299966 697127 > 3²²¹ [i]