MinT - Best Known (189−67, 189, s)-Nets in Base 3

Best Known (189−67, 189, s)-Nets in Base 3

(189−67, 189, 156)-Net over F₃ — Constructive and digital

Digital (122, 189, 156)-net over F₃, using

11 times m-reduction [i] based on digital (122, 200, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 100, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(189−67, 189, 252)-Net over F₃ — Digital

Digital (122, 189, 252)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−67, 189, 3406)-Net in Base 3 — Upper bound on s

There is no (122, 189, 3407)-net in base 3, because

1 times m-reduction [i] would yield (122, 188, 3407)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 500170 064938 653096 438959 663832 260123 384029 375406 976498 896526 366829 283478 632847 185384 819295 > 3¹⁸⁸ [i]