MinT - Best Known (69−45, 69, s)-Nets in Base 7

Best Known (69−45, 69, s)-Nets in Base 7

(69−45, 69, 30)-Net over F₇ — Constructive and digital

Digital (24, 69, 30)-net over F₇, using

t-expansion [i] based on digital (23, 69, 30)-net over F₇, using
- net from sequence [i] based on digital (23, 29)-sequence over F₇, using
  - Niederreiter sequence [i]

(69−45, 69, 72)-Net over F₇ — Digital

Digital (24, 69, 72)-net over F₇, using

t-expansion [i] based on digital (22, 69, 72)-net over F₇, using
- net from sequence [i] based on digital (22, 71)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 22 and N(F) ≥ 72, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(69−45, 69, 603)-Net in Base 7 — Upper bound on s

There is no (24, 69, 604)-net in base 7, because

1 times m-reduction [i] would yield (24, 68, 604)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2959 871920 408097 946574 546857 299070 448428 606357 970073 448649 > 7⁶⁸ [i]