MinT - Best Known (42, 64, s)-Nets in Base 7

Best Known (42, 64, s)-Nets in Base 7

(42, 64, 129)-Net over F₇ — Constructive and digital

Digital (42, 64, 129)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 20, 29)-net over F₇, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 5, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)
    3. digital (1, 12, 13)-net over F₇, using
      - 1 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
2. digital (22, 44, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 22, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]

(42, 64, 555)-Net over F₇ — Digital

Digital (42, 64, 555)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 64, 67569)-Net in Base 7 — Upper bound on s

There is no (42, 64, 67570)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 219787 628430 243099 664782 143403 203531 603000 383210 605369 > 7⁶⁴ [i]