MinT - Best Known (37−17, 37, s)-Nets in Base 16

Best Known (37−17, 37, s)-Nets in Base 16

(37−17, 37, 516)-Net over F₁₆ — Constructive and digital

Digital (20, 37, 516)-net over F₁₆, using

1 times m-reduction [i] based on digital (20, 38, 516)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 19, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(37−17, 37, 578)-Net over F₁₆ — Digital

Digital (20, 37, 578)-net over F₁₆, using

1 times m-reduction [i] based on digital (20, 38, 578)-net over F₁₆, using
- trace code for nets [i] based on digital (1, 19, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(37−17, 37, 65782)-Net in Base 16 — Upper bound on s

There is no (20, 37, 65783)-net in base 16, because

1 times m-reduction [i] would yield (20, 36, 65783)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 22 301391 609128 932636 829986 660386 122349 348886 > 16³⁶ [i]