MinT - Best Known (51−17, 51, s)-Nets in Base 7

Best Known (51−17, 51, s)-Nets in Base 7

(51−17, 51, 200)-Net over F₇ — Constructive and digital

Digital (34, 51, 200)-net over F₇, using

7¹ times duplication [i] based on digital (33, 50, 200)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 16, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 8, 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]
  2. digital (17, 34, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 17, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(51−17, 51, 568)-Net over F₇ — Digital

Digital (34, 51, 568)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−17, 51, 120056)-Net in Base 7 — Upper bound on s

There is no (34, 51, 120057)-net in base 7, because

1 times m-reduction [i] would yield (34, 50, 120057)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 798577 861183 739606 352890 009905 362191 893185 > 7⁵⁰ [i]