MinT - Best Known (19−12, 19, s)-Nets in Base 7

Best Known (19−12, 19, s)-Nets in Base 7

(19−12, 19, 21)-Net over F₇ — Constructive and digital

Digital (7, 19, 21)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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 (1, 13, 13)-net over F₇, using
  - a shift-net [i]

(19−12, 19, 32)-Net over F₇ — Digital

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]

(19−12, 19, 233)-Net in Base 7 — Upper bound on s

There is no (7, 19, 234)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 11623 998787 279165 > 7¹⁹ [i]