MinT - Best Known (51−33, 51, s)-Nets in Base 81

Best Known (51−33, 51, s)-Nets in Base 81

(51−33, 51, 370)-Net over F₈₁ — Constructive and digital

Digital (18, 51, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 51, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(51−33, 51, 78238)-Net in Base 81 — Upper bound on s

There is no (18, 51, 78239)-net in base 81, because

1 times m-reduction [i] would yield (18, 50, 78239)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 265629 725156 968672 835863 765879 128684 855437 285285 364197 349627 085491 122339 454084 615584 962516 785921 > 81⁵⁰ [i]