MinT - Best Known (57−25, 57, s)-Nets in Base 3

Best Known (57−25, 57, s)-Nets in Base 3

(57−25, 57, 56)-Net over F₃ — Constructive and digital

Digital (32, 57, 56)-net over F₃, using

1 times m-reduction [i] based on digital (32, 58, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 29, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]

(57−25, 57, 434)-Net in Base 3 — Upper bound on s

There is no (32, 57, 435)-net in base 3, because

1 times m-reduction [i] would yield (32, 56, 435)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 535 117457 537167 771864 349001 > 3⁵⁶ [i]