MinT - Best Known (165, 165+84, s)-Nets in Base 3

Best Known (165, 165+84, s)-Nets in Base 3

(165, 165+84, 162)-Net over F₃ — Constructive and digital

Digital (165, 249, 162)-net over F₃, using

t-expansion [i] based on digital (157, 249, 162)-net over F₃, using
- 1 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]

(165, 165+84, 371)-Net over F₃ — Digital

Digital (165, 249, 371)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+84, 5523)-Net in Base 3 — Upper bound on s

There is no (165, 249, 5524)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64022 956953 925816 841526 381454 646545 042307 276169 882882 316291 040509 111670 728119 761334 653430 037742 182116 292236 489483 662361 > 3²⁴⁹ [i]