MinT - Best Known (50, 50+45, s)-Nets in Base 16

Best Known (50, 50+45, s)-Nets in Base 16

(50, 50+45, 518)-Net over F₁₆ — Constructive and digital

Digital (50, 95, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (50, 96, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 48, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(50, 50+45, 642)-Net over F₁₆ — Digital

Digital (50, 95, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (50, 96, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 48, 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]

(50, 50+45, 84251)-Net in Base 16 — Upper bound on s

There is no (50, 95, 84252)-net in base 16, because

1 times m-reduction [i] would yield (50, 94, 84252)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 153944 027318 923625 446225 366131 733655 885477 837422 405588 850044 577552 260401 530306 984442 620178 701912 822902 909544 286586 > 16⁹⁴ [i]