MinT - Best Known (164, 164+83, s)-Nets in Base 3

Best Known (164, 164+83, s)-Nets in Base 3

(164, 164+83, 162)-Net over F₃ — Constructive and digital

Digital (164, 247, 162)-net over F₃, using

t-expansion [i] based on digital (157, 247, 162)-net over F₃, using
- 3 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
  - trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
    - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
        F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(164, 164+83, 373)-Net over F₃ — Digital

Digital (164, 247, 373)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+83, 5842)-Net in Base 3 — Upper bound on s

There is no (164, 247, 5843)-net in base 3, because

1 times m-reduction [i] would yield (164, 246, 5843)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2357 848976 312104 826402 588173 368819 554964 721841 518633 658964 749211 606966 925151 535605 063050 605669 017597 882260 700574 825495 > 3²⁴⁶ [i]