MinT - Best Known (163, 163+85, s)-Nets in Base 3

Best Known (163, 163+85, s)-Nets in Base 3

(163, 163+85, 162)-Net over F₃ — Constructive and digital

Digital (163, 248, 162)-net over F₃, using

t-expansion [i] based on 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]

(163, 163+85, 353)-Net over F₃ — Digital

Digital (163, 248, 353)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+85, 5239)-Net in Base 3 — Upper bound on s

There is no (163, 248, 5240)-net in base 3, because

1 times m-reduction [i] would yield (163, 247, 5240)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7094 459844 488094 556559 817418 453703 971725 597077 101338 371042 097176 662727 354926 144547 308220 775691 452937 306537 904685 037521 > 3²⁴⁷ [i]