MinT - Best Known (185−69, 185, s)-Nets in Base 3

Best Known (185−69, 185, s)-Nets in Base 3

(185−69, 185, 156)-Net over F₃ — Constructive and digital

Digital (116, 185, 156)-net over F₃, using

3 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]

(185−69, 185, 213)-Net over F₃ — Digital

Digital (116, 185, 213)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(185−69, 185, 2551)-Net in Base 3 — Upper bound on s

There is no (116, 185, 2552)-net in base 3, because

1 times m-reduction [i] would yield (116, 184, 2552)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6170 799743 035474 716887 949131 676969 789142 651444 970014 184842 180373 933330 752538 415519 853137 > 3¹⁸⁴ [i]