MinT - Best Known (108, 165, s)-Nets in Base 3

Best Known (108, 165, s)-Nets in Base 3

(108, 165, 156)-Net over F₃ — Constructive and digital

Digital (108, 165, 156)-net over F₃, using

7 times m-reduction [i] based on digital (108, 172, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 86, 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]

(108, 165, 242)-Net over F₃ — Digital

Digital (108, 165, 242)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(108, 165, 3492)-Net in Base 3 — Upper bound on s

There is no (108, 165, 3493)-net in base 3, because

1 times m-reduction [i] would yield (108, 164, 3493)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 775464 533726 517655 986435 867901 502725 352758 534799 911657 644286 801672 561554 703601 > 3¹⁶⁴ [i]