MinT - Best Known (39, 39+21, s)-Nets in Base 7

Best Known (39, 39+21, s)-Nets in Base 7

(39, 39+21, 126)-Net over F₇ — Constructive and digital

Digital (39, 60, 126)-net over F₇, using

7¹ times duplication [i] based on digital (38, 59, 126)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 17, 26)-net over F₇, using
    - (u,Â u+v)-construction [i] based on
      1. digital (1, 6, 13)-net over F₇, using
        7 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
        a shift-net [i]
      2. digital (1, 11, 13)-net over F₇, using
        2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇ (see above)
  2. digital (21, 42, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 21, 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]

(39, 39+21, 485)-Net over F₇ — Digital

Digital (39, 60, 485)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 39+21, 73091)-Net in Base 7 — Upper bound on s

There is no (39, 60, 73092)-net in base 7, because

1 times m-reduction [i] would yield (39, 59, 73092)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 72 579556 236447 986808 240207 176670 882621 498614 703769 > 7⁵⁹ [i]