MinT - Best Known (89−67, 89, s)-Nets in Base 7

Best Known (89−67, 89, s)-Nets in Base 7

(89−67, 89, 29)-Net over F₇ — Constructive and digital

Digital (22, 89, 29)-net over F₇, using

t-expansion [i] based on digital (21, 89, 29)-net over F₇, using
- net from sequence [i] based on digital (21, 28)-sequence over F₇, using
  - Niederreiter sequence [i]

(89−67, 89, 72)-Net over F₇ — Digital

Digital (22, 89, 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]

(89−67, 89, 372)-Net in Base 7 — Upper bound on s

There is no (22, 89, 373)-net in base 7, because

1 times m-reduction [i] would yield (22, 88, 373)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 249 186436 548063 306830 061216 808786 984973 597668 828607 619840 441106 217252 609983 > 7⁸⁸ [i]