MinT - Best Known (3, 3+7, s)-Nets in Base 7

Best Known (3, 3+7, s)-Nets in Base 7

(3, 3+7, 16)-Net over F₇ — Constructive and digital

Digital (3, 10, 16)-net over F₇, using

(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, 7, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)

(3, 3+7, 20)-Net over F₇ — Digital

Digital (3, 10, 20)-net over F₇, using

net from sequence [i] based on digital (3, 19)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 3 and N(F) ≥ 20, using
  - fields by Serre with genus 3 or 4 [i]

(3, 3+7, 102)-Net in Base 7 — Upper bound on s

There is no (3, 10, 103)-net in base 7, because

1 times m-reduction [i] would yield (3, 9, 103)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 41 071663 > 7⁹ [i]