MinT - Best Known (221−81, 221, s)-Nets in Base 3

Best Known (221−81, 221, s)-Nets in Base 3

(221−81, 221, 156)-Net over F₃ — Constructive and digital

Digital (140, 221, 156)-net over F₃, using

15 times m-reduction [i] based on digital (140, 236, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 118, 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]

(221−81, 221, 262)-Net over F₃ — Digital

Digital (140, 221, 262)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−81, 221, 3279)-Net in Base 3 — Upper bound on s

There is no (140, 221, 3280)-net in base 3, because

1 times m-reduction [i] would yield (140, 220, 3280)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 936 356536 530578 382444 052855 802216 680536 467590 945034 344427 409077 443849 671831 680536 006315 775822 550993 618945 > 3²²⁰ [i]