MinT - Best Known (142−103, 142, s)-Nets in Base 9

Best Known (142−103, 142, s)-Nets in Base 9

(142−103, 142, 81)-Net over F₉ — Constructive and digital

Digital (39, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 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]

(142−103, 142, 140)-Net over F₉ — Digital

Digital (39, 142, 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]

(142−103, 142, 1047)-Net in Base 9 — Upper bound on s

There is no (39, 142, 1048)-net in base 9, because

1 times m-reduction [i] would yield (39, 141, 1048)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 357 542008 315338 256314 421724 911724 029561 375896 835944 078566 945460 896615 960721 796866 151631 144626 260491 561609 011971 651572 786764 713278 037825 > 9¹⁴¹ [i]