MinT - Best Known (180−64, 180, s)-Nets in Base 3

Best Known (180−64, 180, s)-Nets in Base 3

(180−64, 180, 156)-Net over F₃ — Constructive and digital

Digital (116, 180, 156)-net over F₃, using

8 times m-reduction [i] based on digital (116, 188, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 94, 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]

(180−64, 180, 239)-Net over F₃ — Digital

Digital (116, 180, 239)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−64, 180, 3056)-Net in Base 3 — Upper bound on s

There is no (116, 180, 3057)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 76 579204 740180 275682 310853 570928 193835 373962 374485 487677 766105 602571 583865 360294 934145 > 3¹⁸⁰ [i]