MinT - Best Known (23−13, 23, s)-Nets in Base 7

Best Known (23−13, 23, s)-Nets in Base 7

(23−13, 23, 24)-Net over F₇ — Constructive and digital

Digital (10, 23, 24)-net over F₇, using

generalized (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 (0, 6, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)
3. digital (0, 13, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)

(23−13, 23, 38)-Net over F₇ — Digital

Digital (10, 23, 38)-net over F₇, using

t-expansion [i] based on digital (9, 23, 38)-net over F₇, using
- net from sequence [i] based on digital (9, 37)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(23−13, 23, 622)-Net in Base 7 — Upper bound on s

There is no (10, 23, 623)-net in base 7, because

1 times m-reduction [i] would yield (10, 22, 623)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 918072 492701 793469 > 7²² [i]