MinT - Best Known (157, 157+91, s)-Nets in Base 3

Best Known (157, 157+91, s)-Nets in Base 3

(157, 157+91, 162)-Net over F₃ — Constructive and digital

Digital (157, 248, 162)-net over F₃, using

2 times m-reduction [i] based on digital (157, 250, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 125, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(157, 157+91, 289)-Net over F₃ — Digital

Digital (157, 248, 289)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+91, 3620)-Net in Base 3 — Upper bound on s

There is no (157, 248, 3621)-net in base 3, because

1 times m-reduction [i] would yield (157, 247, 3621)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7134 089913 672015 691852 790200 704923 361855 379024 254820 995381 628995 633423 304295 676779 301440 978357 461738 284726 935123 923131 > 3²⁴⁷ [i]