MinT - Best Known (168−65, 168, s)-Nets in Base 3

Best Known (168−65, 168, s)-Nets in Base 3

(168−65, 168, 148)-Net over F₃ — Constructive and digital

Digital (103, 168, 148)-net over F₃, using

4 times m-reduction [i] based on digital (103, 172, 148)-net over F₃, using
- trace code for nets [i] based on digital (17, 86, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(168−65, 168, 177)-Net over F₃ — Digital

Digital (103, 168, 177)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−65, 168, 1944)-Net in Base 3 — Upper bound on s

There is no (103, 168, 1945)-net in base 3, because

1 times m-reduction [i] would yield (103, 167, 1945)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47 865682 478009 789919 584233 358249 350695 817227 153487 704434 344124 380731 731120 070273 > 3¹⁶⁷ [i]