MinT - Best Known (54, 54+31, s)-Nets in Base 3

Best Known (54, 54+31, s)-Nets in Base 3

(54, 54+31, 80)-Net over F₃ — Constructive and digital

Digital (54, 85, 80)-net over F₃, using

7 times m-reduction [i] based on digital (54, 92, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 46, 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]

(54, 54+31, 122)-Net over F₃ — Digital

Digital (54, 85, 122)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+31, 1494)-Net in Base 3 — Upper bound on s

There is no (54, 85, 1495)-net in base 3, because

1 times m-reduction [i] would yield (54, 84, 1495)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12055 813581 933504 214604 667056 527861 100355 > 3⁸⁴ [i]