MinT - Best Known (19−11, 19, s)-Nets in Base 7

Best Known (19−11, 19, s)-Nets in Base 7

(19−11, 19, 26)-Net over F₇ — Constructive and digital

Digital (8, 19, 26)-net over F₇, using

1 times m-reduction [i] based on digital (8, 20, 26)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 7, 13)-net over F₇, using
    - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
      - a shift-net [i]
  2. digital (1, 13, 13)-net over F₇ (see above)

(19−11, 19, 32)-Net over F₇ — Digital

Digital (8, 19, 32)-net over F₇, using

t-expansion [i] based on digital (7, 19, 32)-net over F₇, using
- net from sequence [i] based on digital (7, 31)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 7 and N(F) ≥ 32, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(19−11, 19, 475)-Net in Base 7 — Upper bound on s

There is no (8, 19, 476)-net in base 7, because

1 times m-reduction [i] would yield (8, 18, 476)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1631 056661 464777 > 7¹⁸ [i]