MinT - Best Known (245−99, 245, s)-Nets in Base 3

Best Known (245−99, 245, s)-Nets in Base 3

(245−99, 245, 156)-Net over F₃ — Constructive and digital

Digital (146, 245, 156)-net over F₃, using

3 times m-reduction [i] based on digital (146, 248, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 124, 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]

(245−99, 245, 217)-Net over F₃ — Digital

Digital (146, 245, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−99, 245, 2222)-Net in Base 3 — Upper bound on s

There is no (146, 245, 2223)-net in base 3, because

1 times m-reduction [i] would yield (146, 244, 2223)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 264 050594 157979 824616 639519 047886 183218 518535 912316 314236 484142 230923 281900 292533 321279 148517 549945 781328 644398 821631 > 3²⁴⁴ [i]