MinT - Best Known (40, 40+71, s)-Nets in Base 9

Best Known (40, 40+71, s)-Nets in Base 9

(40, 40+71, 81)-Net over F₉ — Constructive and digital

Digital (40, 111, 81)-net over F₉, using

t-expansion [i] based on digital (32, 111, 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]

(40, 40+71, 140)-Net over F₉ — Digital

Digital (40, 111, 140)-net over F₉, using

t-expansion [i] based on digital (39, 111, 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]

(40, 40+71, 1713)-Net in Base 9 — Upper bound on s

There is no (40, 111, 1714)-net in base 9, because

1 times m-reduction [i] would yield (40, 110, 1714)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 937 577562 241135 838728 248050 003924 871483 639774 924843 948276 875089 927193 012842 993930 881040 422999 794127 078129 > 9¹¹⁰ [i]