MinT - Best Known (107−51, 107, s)-Nets in Base 16

Best Known (107−51, 107, s)-Nets in Base 16

(107−51, 107, 518)-Net over F₁₆ — Constructive and digital

Digital (56, 107, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 108, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 54, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(107−51, 107, 642)-Net over F₁₆ — Digital

Digital (56, 107, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 108, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 54, 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]

(107−51, 107, 86483)-Net in Base 16 — Upper bound on s

There is no (56, 107, 86484)-net in base 16, because

1 times m-reduction [i] would yield (56, 106, 86484)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 325694 664651 127233 484298 340131 544977 428659 006327 398097 080781 634571 667823 455937 998029 326308 711991 395629 545036 770456 734276 828376 > 16¹⁰⁶ [i]