MinT - Best Known (9, 20, s)-Nets in Base 7

Best Known (9, 20, s)-Nets in Base 7

(9, 20, 29)-Net over F₇ — Constructive and digital

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]

(9, 20, 39)-Net over F₇ — Digital

Digital (9, 20, 39)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(9, 20, 703)-Net in Base 7 — Upper bound on s

There is no (9, 20, 704)-net in base 7, because

1 times m-reduction [i] would yield (9, 19, 704)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 11432 661225 254017 > 7¹⁹ [i]