MinT - Best Known (23−15, 23, s)-Nets in Base 7

Best Known (23−15, 23, s)-Nets in Base 7

(23−15, 23, 17)-Net over F₇ — Constructive and digital

Digital (8, 23, 17)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 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, 16, 9)-net over F₇, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₇, using
    - Niederreiter sequence [i]

(23−15, 23, 32)-Net over F₇ — Digital

Digital (8, 23, 32)-net over F₇, using

t-expansion [i] based on digital (7, 23, 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]

(23−15, 23, 251)-Net in Base 7 — Upper bound on s

There is no (8, 23, 252)-net in base 7, because

1 times m-reduction [i] would yield (8, 22, 252)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4 017966 350448 391201 > 7²² [i]