MinT - Best Known (54, 54+45, s)-Nets in Base 16

Best Known (54, 54+45, s)-Nets in Base 16

(54, 54+45, 522)-Net over F₁₆ — Constructive and digital

Digital (54, 99, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (54, 100, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 50, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(54, 54+45, 642)-Net over F₁₆ — Digital

Digital (54, 99, 642)-net over F₁₆, using

5 times m-reduction [i] based on digital (54, 104, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 52, 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]

(54, 54+45, 139486)-Net in Base 16 — Upper bound on s

There is no (54, 99, 139487)-net in base 16, because

1 times m-reduction [i] would yield (54, 98, 139487)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10087 949095 248640 486315 670540 548163 109984 878607 802240 085349 245503 167105 935309 738779 064801 977440 628369 352446 426898 399761 > 16⁹⁸ [i]