MinT - Best Known (59−27, 59, s)-Nets in Base 16

Best Known (59−27, 59, s)-Nets in Base 16

(59−27, 59, 518)-Net over F₁₆ — Constructive and digital

Digital (32, 59, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (32, 60, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 30, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(59−27, 59, 642)-Net over F₁₆ — Digital

Digital (32, 59, 642)-net over F₁₆, using

1 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]

(59−27, 59, 89024)-Net in Base 16 — Upper bound on s

There is no (32, 59, 89025)-net in base 16, because

1 times m-reduction [i] would yield (32, 58, 89025)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6902 354642 057909 716885 201098 230835 281847 547028 535115 534874 989674 957376 > 16⁵⁸ [i]