MinT - Best Known (40, 40+21, s)-Nets in Base 3

Best Known (40, 40+21, s)-Nets in Base 3

(40, 40+21, 80)-Net over F₃ — Constructive and digital

Digital (40, 61, 80)-net over F₃, using

3 times m-reduction [i] based on digital (40, 64, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 32, 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]

(40, 40+21, 114)-Net over F₃ — Digital

Digital (40, 61, 114)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+21, 1641)-Net in Base 3 — Upper bound on s

There is no (40, 61, 1642)-net in base 3, because

1 times m-reduction [i] would yield (40, 60, 1642)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 42585 580027 803637 867137 150357 > 3⁶⁰ [i]