MinT - Best Known (189, 189+55, s)-Nets in Base 3

Best Known (189, 189+55, s)-Nets in Base 3

(189, 189+55, 640)-Net over F₃ — Constructive and digital

Digital (189, 244, 640)-net over F₃, using

t-expansion [i] based on digital (188, 244, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 61, 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]

(189, 189+55, 1528)-Net over F₃ — Digital

Digital (189, 244, 1528)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189, 189+55, 107488)-Net in Base 3 — Upper bound on s

There is no (189, 244, 107489)-net in base 3, because

1 times m-reduction [i] would yield (189, 243, 107489)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 205911 463129 798696 015832 348917 722431 456519 688093 020855 438532 347523 572669 180174 069122 142519 457905 117875 116145 374267 > 3²⁴³ [i]