MinT - Best Known (51−29, 51, s)-Nets in Base 7

Best Known (51−29, 51, s)-Nets in Base 7

(51−29, 51, 29)-Net over F₇ — Constructive and digital

Digital (22, 51, 29)-net over F₇, using

t-expansion [i] based on digital (21, 51, 29)-net over F₇, using
- net from sequence [i] based on digital (21, 28)-sequence over F₇, using
  - Niederreiter sequence [i]

(51−29, 51, 72)-Net over F₇ — Digital

Digital (22, 51, 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]

(51−29, 51, 1041)-Net in Base 7 — Upper bound on s

There is no (22, 51, 1042)-net in base 7, because

1 times m-reduction [i] would yield (22, 50, 1042)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1 799041 750333 325753 137910 507803 181027 246077 > 7⁵⁰ [i]