MinT - Best Known (40−31, 40, s)-Nets in Base 16

Best Known (40−31, 40, s)-Nets in Base 16

(40−31, 40, 65)-Net over F₁₆ — Constructive and digital

Digital (9, 40, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 40, 65)-net over F₁₆, using
- net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
    - the Hermitian function field over F₁₆ [i]

(40−31, 40, 72)-Net over F₁₆ — Digital

Digital (9, 40, 72)-net over F₁₆, using

net from sequence [i] based on digital (9, 71)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 9 and N(F) ≥ 72, using
  - a curve from the manYPoints database [i]

(40−31, 40, 570)-Net in Base 16 — Upper bound on s

There is no (9, 40, 571)-net in base 16, because

1 times m-reduction [i] would yield (9, 39, 571)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 92198 204706 102670 187688 018302 329804 825019 208976 > 16³⁹ [i]