MinT - Best Known (186−41, 186, s)-Nets in Base 3

Best Known (186−41, 186, s)-Nets in Base 3

(186−41, 186, 640)-Net over F₃ — Constructive and digital

Digital (145, 186, 640)-net over F₃, using

3² times duplication [i] based on digital (143, 184, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 46, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(186−41, 186, 1324)-Net over F₃ — Digital

Digital (145, 186, 1324)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−41, 186, 107539)-Net in Base 3 — Upper bound on s

There is no (145, 186, 107540)-net in base 3, because

1 times m-reduction [i] would yield (145, 185, 107540)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18511 886161 309600 861152 918268 808629 376285 743723 950472 179302 272533 212361 654508 429179 988929 > 3¹⁸⁵ [i]