MinT - Best Known (53−23, 53, s)-Nets in Base 16

Best Known (53−23, 53, s)-Nets in Base 16

(53−23, 53, 520)-Net over F₁₆ — Constructive and digital

Digital (30, 53, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (30, 54, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 27, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(53−23, 53, 642)-Net over F₁₆ — Digital

Digital (30, 53, 642)-net over F₁₆, using

3 times m-reduction [i] based on digital (30, 56, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 28, 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]

(53−23, 53, 161105)-Net in Base 16 — Upper bound on s

There is no (30, 53, 161106)-net in base 16, because

1 times m-reduction [i] would yield (30, 52, 161106)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 411 383624 941798 385749 843792 992451 819473 322625 195208 041917 908616 > 16⁵² [i]