MinT - Best Known (189−81, 189, s)-Nets in Base 3

Best Known (189−81, 189, s)-Nets in Base 3

(189−81, 189, 128)-Net over F₃ — Constructive and digital

Digital (108, 189, 128)-net over F₃, using

1 times m-reduction [i] based on digital (108, 190, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 95, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(189−81, 189, 147)-Net over F₃ — Digital

Digital (108, 189, 147)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−81, 189, 1338)-Net in Base 3 — Upper bound on s

There is no (108, 189, 1339)-net in base 3, because

1 times m-reduction [i] would yield (108, 188, 1339)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 501142 484966 229611 006235 896488 650334 406095 819916 970714 539935 687116 042532 574562 449026 791729 > 3¹⁸⁸ [i]