MinT - Best Known (81−37, 81, s)-Nets in Base 16

Best Known (81−37, 81, s)-Nets in Base 16

(81−37, 81, 520)-Net over F₁₆ — Constructive and digital

Digital (44, 81, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 82, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 41, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(81−37, 81, 642)-Net over F₁₆ — Digital

Digital (44, 81, 642)-net over F₁₆, using

3 times m-reduction [i] based on digital (44, 84, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 42, 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]

(81−37, 81, 113147)-Net in Base 16 — Upper bound on s

There is no (44, 81, 113148)-net in base 16, because

1 times m-reduction [i] would yield (44, 80, 113148)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136070 765080 053795 804866 784169 252832 483965 063669 968084 060000 907897 275289 203686 739849 173604 345011 > 16⁸⁰ [i]