MinT - Best Known (35−11, 35, s)-Nets in Base 7

Best Known (35−11, 35, s)-Nets in Base 7

(35−11, 35, 208)-Net over F₇ — Constructive and digital

Digital (24, 35, 208)-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 (5, 10, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 5, 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]
3. digital (11, 22, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 11, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(35−11, 35, 690)-Net over F₇ — Digital

Digital (24, 35, 690)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(35−11, 35, 242296)-Net in Base 7 — Upper bound on s

There is no (24, 35, 242297)-net in base 7, because

1 times m-reduction [i] would yield (24, 34, 242297)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 54117 856009 661654 721757 033591 > 7³⁴ [i]