MinT - Best Known (57−25, 57, s)-Nets in Base 16

Best Known (57−25, 57, s)-Nets in Base 16

(57−25, 57, 520)-Net over F₁₆ — Constructive and digital

Digital (32, 57, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (32, 58, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 29, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(57−25, 57, 642)-Net over F₁₆ — Digital

Digital (32, 57, 642)-net over F₁₆, using

3 times m-reduction [i] based on digital (32, 60, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 30, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(57−25, 57, 146716)-Net in Base 16 — Upper bound on s

There is no (32, 57, 146717)-net in base 16, because

1 times m-reduction [i] would yield (32, 56, 146717)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 26 961694 130071 742988 741061 933559 703859 426477 690389 533522 190107 963611 > 16⁵⁶ [i]