MinT - Best Known (40, 40+33, s)-Nets in Base 3

Best Known (40, 40+33, s)-Nets in Base 3

(40, 40+33, 56)-Net over F₃ — Constructive and digital

Digital (40, 73, 56)-net over F₃, using

1 times m-reduction [i] based on digital (40, 74, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 37, 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]

(40, 40+33, 59)-Net over F₃ — Digital

Digital (40, 73, 59)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+33, 461)-Net in Base 3 — Upper bound on s

There is no (40, 73, 462)-net in base 3, because

1 times m-reduction [i] would yield (40, 72, 462)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22720 497877 169658 589475 179405 414305 > 3⁷² [i]