MinT - Best Known (224−86, 224, s)-Nets in Base 3

Best Known (224−86, 224, s)-Nets in Base 3

(224−86, 224, 156)-Net over F₃ — Constructive and digital

Digital (138, 224, 156)-net over F₃, using

8 times m-reduction [i] based on digital (138, 232, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 116, 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]

(224−86, 224, 232)-Net over F₃ — Digital

Digital (138, 224, 232)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−86, 224, 2539)-Net in Base 3 — Upper bound on s

There is no (138, 224, 2540)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75645 791884 816467 349055 745149 260417 834150 536240 055472 814393 297010 091157 160629 409016 141927 120536 931606 320785 > 3²²⁴ [i]