MinT - Best Known (165, 165+67, s)-Nets in Base 3

Best Known (165, 165+67, s)-Nets in Base 3

(165, 165+67, 282)-Net over F₃ — Constructive and digital

Digital (165, 232, 282)-net over F₃, using

3¹ times duplication [i] based on digital (164, 231, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 77, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(165, 165+67, 571)-Net over F₃ — Digital

Digital (165, 232, 571)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+67, 14361)-Net in Base 3 — Upper bound on s

There is no (165, 232, 14362)-net in base 3, because

1 times m-reduction [i] would yield (165, 231, 14362)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 164 346688 023278 527737 184380 491031 369852 229672 644446 540713 964854 874058 187593 355232 432428 273224 420801 607859 633397 > 3²³¹ [i]