MinT - Best Known (42, 42+31, s)-Nets in Base 3

Best Known (42, 42+31, s)-Nets in Base 3

(42, 42+31, 64)-Net over F₃ — Constructive and digital

Digital (42, 73, 64)-net over F₃, using

1 times m-reduction [i] based on digital (42, 74, 64)-net over F₃, using
- trace code for nets [i] based on digital (5, 37, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]

(42, 42+31, 70)-Net over F₃ — Digital

Digital (42, 73, 70)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+31, 612)-Net in Base 3 — Upper bound on s

There is no (42, 73, 613)-net in base 3, because

1 times m-reduction [i] would yield (42, 72, 613)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22999 381337 875004 160962 921104 656891 > 3⁷² [i]