MinT - Best Known (13−8, 13, s)-Nets in Base 7

Best Known (13−8, 13, s)-Nets in Base 7

(13−8, 13, 21)-Net over F₇ — Constructive and digital

Digital (5, 13, 21)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 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, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]

(13−8, 13, 24)-Net over F₇ — Digital

Digital (5, 13, 24)-net over F₇, using

t-expansion [i] based on digital (4, 13, 24)-net over F₇, using
- net from sequence [i] based on digital (4, 23)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 4 and N(F) ≥ 24, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(13−8, 13, 203)-Net in Base 7 — Upper bound on s

There is no (5, 13, 204)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 97542 645697 > 7¹³ [i]