MinT - Best Known (189−49, 189, s)-Nets in Base 3

Best Known (189−49, 189, s)-Nets in Base 3

(189−49, 189, 288)-Net over F₃ — Constructive and digital

Digital (140, 189, 288)-net over F₃, using

t-expansion [i] based on digital (139, 189, 288)-net over F₃, using
- 3 times m-reduction [i] based on digital (139, 192, 288)-net over F₃, using
  - trace code for nets [i] based on digital (11, 64, 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]

(189−49, 189, 704)-Net over F₃ — Digital

Digital (140, 189, 704)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−49, 189, 26754)-Net in Base 3 — Upper bound on s

There is no (140, 189, 26755)-net in base 3, because

1 times m-reduction [i] would yield (140, 188, 26755)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 499897 289445 695460 240917 874163 823001 519820 266110 256532 616422 674094 189228 996796 696706 517201 > 3¹⁸⁸ [i]