MinT - Best Known (89−55, 89, s)-Nets in Base 7

Best Known (89−55, 89, s)-Nets in Base 7

(89−55, 89, 35)-Net over F₇ — Constructive and digital

Digital (34, 89, 35)-net over F₇, using

t-expansion [i] based on digital (33, 89, 35)-net over F₇, using
- net from sequence [i] based on digital (33, 34)-sequence over F₇, using
  - Niederreiter sequence [i]

(89−55, 89, 96)-Net over F₇ — Digital

Digital (34, 89, 96)-net over F₇, using

t-expansion [i] based on digital (33, 89, 96)-net over F₇, using
- net from sequence [i] based on digital (33, 95)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 33 and N(F) ≥ 96, using
    - an algebraic function field by Teo [i]

(89−55, 89, 1016)-Net in Base 7 — Upper bound on s

There is no (34, 89, 1017)-net in base 7, because

1 times m-reduction [i] would yield (34, 88, 1017)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 234 289019 908863 354572 043663 781327 779425 190763 644232 891811 941656 689200 907671 > 7⁸⁸ [i]