MinT - Best Known (19−13, 19, s)-Nets in Base 81

Best Known (19−13, 19, s)-Nets in Base 81

(19−13, 19, 164)-Net over F₈₁ — Constructive and digital

Digital (6, 19, 164)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 13, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(19−13, 19, 190)-Net over F₈₁ — Digital

Digital (6, 19, 190)-net over F₈₁, using

net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
  - a curve from the manYPoints database [i]

(19−13, 19, 19885)-Net in Base 81 — Upper bound on s

There is no (6, 19, 19886)-net in base 81, because

1 times m-reduction [i] would yield (6, 18, 19886)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 22533 539723 477858 084061 884785 321281 > 81¹⁸ [i]