MinT - Best Known (239−89, 239, s)-Nets in Base 3

Best Known (239−89, 239, s)-Nets in Base 3

(239−89, 239, 156)-Net over F₃ — Constructive and digital

Digital (150, 239, 156)-net over F₃, using

t-expansion [i] based on digital (147, 239, 156)-net over F₃, using
- 11 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
  - trace code for nets [i] based on digital (22, 125, 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]

(239−89, 239, 268)-Net over F₃ — Digital

Digital (150, 239, 268)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−89, 239, 3242)-Net in Base 3 — Upper bound on s

There is no (150, 239, 3243)-net in base 3, because

1 times m-reduction [i] would yield (150, 238, 3243)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 359428 915773 833069 789790 881559 128680 502301 430602 321056 552314 036773 511433 975905 188371 226789 988698 409768 326143 546505 > 3²³⁸ [i]