MinT - Best Known (67−45, 67, s)-Nets in Base 7

Best Known (67−45, 67, s)-Nets in Base 7

(67−45, 67, 29)-Net over F₇ — Constructive and digital

Digital (22, 67, 29)-net over F₇, using

t-expansion [i] based on digital (21, 67, 29)-net over F₇, using
- net from sequence [i] based on digital (21, 28)-sequence over F₇, using
  - Niederreiter sequence [i]

(67−45, 67, 72)-Net over F₇ — Digital

Digital (22, 67, 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]

(67−45, 67, 503)-Net in Base 7 — Upper bound on s

There is no (22, 67, 504)-net in base 7, because

1 times m-reduction [i] would yield (22, 66, 504)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 60 897679 670211 168982 226780 902557 612979 394080 370261 075089 > 7⁶⁶ [i]