MinT - Best Known (241−89, 241, s)-Nets in Base 3

Best Known (241−89, 241, s)-Nets in Base 3

(241−89, 241, 156)-Net over F₃ — Constructive and digital

Digital (152, 241, 156)-net over F₃, using

t-expansion [i] based on digital (147, 241, 156)-net over F₃, using
- 9 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]

(241−89, 241, 277)-Net over F₃ — Digital

Digital (152, 241, 277)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−89, 241, 3411)-Net in Base 3 — Upper bound on s

There is no (152, 241, 3412)-net in base 3, because

1 times m-reduction [i] would yield (152, 240, 3412)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 265460 467818 317846 669175 636797 740919 923728 795226 519751 964990 257658 993755 370510 312110 095043 311854 300575 520016 448385 > 3²⁴⁰ [i]