MinT - Best Known (40, 40+31, s)-Nets in Base 3

Best Known (40, 40+31, s)-Nets in Base 3

(40, 40+31, 60)-Net over F₃ — Constructive and digital

Digital (40, 71, 60)-net over F₃, using

1 times m-reduction [i] based on digital (40, 72, 60)-net over F₃, using
- trace code for nets [i] based on digital (4, 36, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]

(40, 40+31, 64)-Net over F₃ — Digital

Digital (40, 71, 64)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+31, 526)-Net in Base 3 — Upper bound on s

There is no (40, 71, 527)-net in base 3, because

1 times m-reduction [i] would yield (40, 70, 527)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2517 658459 547076 390262 574517 501987 > 3⁷⁰ [i]