MinT - Best Known (165, 165+61, s)-Nets in Base 3

Best Known (165, 165+61, s)-Nets in Base 3

(165, 165+61, 288)-Net over F₃ — Constructive and digital

Digital (165, 226, 288)-net over F₃, using

5 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+61, 705)-Net over F₃ — Digital

Digital (165, 226, 705)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+61, 22782)-Net in Base 3 — Upper bound on s

There is no (165, 226, 22783)-net in base 3, because

1 times m-reduction [i] would yield (165, 225, 22783)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 225121 591308 094786 708928 591460 090066 120742 202370 262676 744576 727009 395599 985986 369806 641227 385693 109595 810245 > 3²²⁵ [i]