MinT - Best Known (225−83, 225, s)-Nets in Base 3

Best Known (225−83, 225, s)-Nets in Base 3

(225−83, 225, 156)-Net over F₃ — Constructive and digital

Digital (142, 225, 156)-net over F₃, using

15 times m-reduction [i] based on digital (142, 240, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 120, 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]

(225−83, 225, 261)-Net over F₃ — Digital

Digital (142, 225, 261)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(225−83, 225, 3222)-Net in Base 3 — Upper bound on s

There is no (142, 225, 3223)-net in base 3, because

1 times m-reduction [i] would yield (142, 224, 3223)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75455 682735 007875 608292 341347 830552 719825 992717 313571 816266 173801 459635 072604 268218 264362 956936 154431 736543 > 3²²⁴ [i]