MinT - Best Known (57, 108, s)-Nets in Base 16

Best Known (57, 108, s)-Nets in Base 16

(57, 108, 520)-Net over F₁₆ — Constructive and digital

Digital (57, 108, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 54, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(57, 108, 642)-Net over F₁₆ — Digital

Digital (57, 108, 642)-net over F₁₆, using

2 times m-reduction [i] based on digital (57, 110, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 55, 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]

(57, 108, 96628)-Net in Base 16 — Upper bound on s

There is no (57, 108, 96629)-net in base 16, because

1 times m-reduction [i] would yield (57, 107, 96629)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 693 199700 512636 175898 863460 900071 852688 756649 411687 176618 424317 032970 109034 579539 616590 564544 624511 878163 760361 040293 276940 723376 > 16¹⁰⁷ [i]