MinT - Best Known (163−69, 163, s)-Nets in Base 3

Best Known (163−69, 163, s)-Nets in Base 3

(163−69, 163, 80)-Net over F₃ — Constructive and digital

Digital (94, 163, 80)-net over F₃, using

9 times m-reduction [i] based on digital (94, 172, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 86, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(163−69, 163, 134)-Net over F₃ — Digital

Digital (94, 163, 134)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−69, 163, 1236)-Net in Base 3 — Upper bound on s

There is no (94, 163, 1237)-net in base 3, because

1 times m-reduction [i] would yield (94, 162, 1237)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 197590 506525 659571 256100 187004 865461 698590 922484 421429 399710 259647 654386 392049 > 3¹⁶² [i]