MinT - Best Known (142−109, 142, s)-Nets in Base 9

Best Known (142−109, 142, s)-Nets in Base 9

(142−109, 142, 81)-Net over F₉ — Constructive and digital

Digital (33, 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]

(142−109, 142, 128)-Net over F₉ — Digital

Digital (33, 142, 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]

(142−109, 142, 780)-Net in Base 9 — Upper bound on s

There is no (33, 142, 781)-net in base 9, because

1 times m-reduction [i] would yield (33, 141, 781)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 374 301109 620911 312759 568676 438663 423813 038955 714399 281674 494223 587048 558480 473415 009760 230552 459388 467577 907694 092623 690985 943393 823025 > 9¹⁴¹ [i]