MinT - Best Known (139, 139+87, s)-Nets in Base 3

Best Known (139, 139+87, s)-Nets in Base 3

(139, 139+87, 156)-Net over F₃ — Constructive and digital

Digital (139, 226, 156)-net over F₃, using

8 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+87, 232)-Net over F₃ — Digital

Digital (139, 226, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+87, 2606)-Net in Base 3 — Upper bound on s

There is no (139, 226, 2607)-net in base 3, because

1 times m-reduction [i] would yield (139, 225, 2607)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 227616 292343 713375 353221 436448 117688 447860 825139 754088 447597 996747 941265 826405 316026 205469 983021 992476 778091 > 3²²⁵ [i]