MinT - Best Known (37, 140, s)-Nets in Base 9

Best Known (37, 140, s)-Nets in Base 9

(37, 140, 81)-Net over F₉ — Constructive and digital

Digital (37, 140, 81)-net over F₉, using

t-expansion [i] based on digital (32, 140, 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]

(37, 140, 128)-Net over F₉ — Digital

Digital (37, 140, 128)-net over F₉, using

t-expansion [i] based on digital (33, 140, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(37, 140, 958)-Net in Base 9 — Upper bound on s

There is no (37, 140, 959)-net in base 9, because

1 times m-reduction [i] would yield (37, 139, 959)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 429050 791445 564230 616399 049668 081655 031439 587330 217115 397448 304111 205721 532946 462932 602582 642978 199309 843793 001021 086536 450340 229225 > 9¹³⁹ [i]