MinT - Best Known (54, 81, s)-Nets in Base 3

Best Known (54, 81, s)-Nets in Base 3

(54, 81, 128)-Net over F₃ — Constructive and digital

Digital (54, 81, 128)-net over F₃, using

1 times m-reduction [i] based on digital (54, 82, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 41, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(54, 81, 154)-Net over F₃ — Digital

Digital (54, 81, 154)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 81, 2433)-Net in Base 3 — Upper bound on s

There is no (54, 81, 2434)-net in base 3, because

1 times m-reduction [i] would yield (54, 80, 2434)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 147 957119 540063 330308 081581 811640 758605 > 3⁸⁰ [i]