MinT - Best Known (139, 139+91, s)-Nets in Base 3

Best Known (139, 139+91, s)-Nets in Base 3

(139, 139+91, 156)-Net over F₃ — Constructive and digital

Digital (139, 230, 156)-net over F₃, using

4 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+91, 218)-Net over F₃ — Digital

Digital (139, 230, 218)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+91, 2317)-Net in Base 3 — Upper bound on s

There is no (139, 230, 2318)-net in base 3, because

1 times m-reduction [i] would yield (139, 229, 2318)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 509935 756258 068894 692930 347063 764305 567731 201601 376576 946824 720288 566205 082090 721700 901715 134879 394691 544421 > 3²²⁹ [i]