MinT - Best Known (165, 165+63, s)-Nets in Base 3

Best Known (165, 165+63, s)-Nets in Base 3

(165, 165+63, 288)-Net over F₃ — Constructive and digital

Digital (165, 228, 288)-net over F₃, using

3 times m-reduction [i] based on digital (165, 231, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 77, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(165, 165+63, 654)-Net over F₃ — Digital

Digital (165, 228, 654)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+63, 19323)-Net in Base 3 — Upper bound on s

There is no (165, 228, 19324)-net in base 3, because

1 times m-reduction [i] would yield (165, 227, 19324)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 028312 110469 510870 506898 691301 613756 545597 841758 538028 607728 052862 710323 713609 094726 156836 525304 978346 234193 > 3²²⁷ [i]