MinT - Best Known (54, 54+29, s)-Nets in Base 3

Best Known (54, 54+29, s)-Nets in Base 3

(54, 54+29, 80)-Net over F₃ — Constructive and digital

Digital (54, 83, 80)-net over F₃, using

9 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+29, 136)-Net over F₃ — Digital

Digital (54, 83, 136)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+29, 1870)-Net in Base 3 — Upper bound on s

There is no (54, 83, 1871)-net in base 3, because

1 times m-reduction [i] would yield (54, 82, 1871)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1338 527511 867597 301797 884591 557709 479941 > 3⁸² [i]