MinT - Best Known (138−99, 138, s)-Nets in Base 9

Best Known (138−99, 138, s)-Nets in Base 9

(138−99, 138, 81)-Net over F₉ — Constructive and digital

Digital (39, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(138−99, 138, 140)-Net over F₉ — Digital

Digital (39, 138, 140)-net over F₉, using

net from sequence [i] based on digital (39, 139)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
  - a curve from the manYPoints database [i]

(138−99, 138, 1082)-Net in Base 9 — Upper bound on s

There is no (39, 138, 1083)-net in base 9, because

1 times m-reduction [i] would yield (39, 137, 1083)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 55391 993948 585681 152866 475877 530312 533559 027546 213976 757973 785297 527794 087902 022526 408828 300325 447835 034286 085487 499500 749108 745817 > 9¹³⁷ [i]