MinT - Best Known (69−31, 69, s)-Nets in Base 3

Best Known (69−31, 69, s)-Nets in Base 3

(69−31, 69, 56)-Net over F₃ — Constructive and digital

Digital (38, 69, 56)-net over F₃, using

1 times m-reduction [i] based on digital (38, 70, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 35, 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]

(69−31, 69, 58)-Net over F₃ — Digital

Digital (38, 69, 58)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−31, 69, 453)-Net in Base 3 — Upper bound on s

There is no (38, 69, 454)-net in base 3, because

1 times m-reduction [i] would yield (38, 68, 454)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 286 431344 391932 919334 089576 547017 > 3⁶⁸ [i]