MinT - Best Known (157, 157+85, s)-Nets in Base 3

Best Known (157, 157+85, s)-Nets in Base 3

(157, 157+85, 162)-Net over F₃ — Constructive and digital

Digital (157, 242, 162)-net over F₃, using

8 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+85, 321)-Net over F₃ — Digital

Digital (157, 242, 321)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+85, 4472)-Net in Base 3 — Upper bound on s

There is no (157, 242, 4473)-net in base 3, because

1 times m-reduction [i] would yield (157, 241, 4473)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9 739729 214351 798367 765874 880744 544519 234400 133150 392622 406916 729401 121606 173844 584185 748872 072273 680155 759749 302233 > 3²⁴¹ [i]