MinT - Best Known (109−51, 109, s)-Nets in Base 16

Best Known (109−51, 109, s)-Nets in Base 16

(109−51, 109, 520)-Net over F₁₆ — Constructive and digital

Digital (58, 109, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 110, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 55, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(109−51, 109, 642)-Net over F₁₆ — Digital

Digital (58, 109, 642)-net over F₁₆, using

3 times m-reduction [i] based on digital (58, 112, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 56, 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]

(109−51, 109, 107963)-Net in Base 16 — Upper bound on s

There is no (58, 109, 107964)-net in base 16, because

1 times m-reduction [i] would yield (58, 108, 107964)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11091 344970 978649 035663 901347 338894 944928 162966 632156 041842 688263 432268 719504 471156 629973 316224 636990 608361 526775 089178 656613 355251 > 16¹⁰⁸ [i]