MinT - Best Known (168−59, 168, s)-Nets in Base 3

Best Known (168−59, 168, s)-Nets in Base 3

(168−59, 168, 156)-Net over F₃ — Constructive and digital

Digital (109, 168, 156)-net over F₃, using

6 times m-reduction [i] based on digital (109, 174, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 87, 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]

(168−59, 168, 234)-Net over F₃ — Digital

Digital (109, 168, 234)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−59, 168, 3234)-Net in Base 3 — Upper bound on s

There is no (109, 168, 3235)-net in base 3, because

1 times m-reduction [i] would yield (109, 167, 3235)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47 836172 888542 909841 216024 076020 407048 090224 871980 929348 429971 452444 197326 902351 > 3¹⁶⁷ [i]