MinT - Best Known (105−49, 105, s)-Nets in Base 16

Best Known (105−49, 105, s)-Nets in Base 16

(105−49, 105, 520)-Net over F₁₆ — Constructive and digital

Digital (56, 105, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (56, 106, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 53, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(105−49, 105, 642)-Net over F₁₆ — Digital

Digital (56, 105, 642)-net over F₁₆, using

3 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]

(105−49, 105, 107912)-Net in Base 16 — Upper bound on s

There is no (56, 105, 107913)-net in base 16, because

1 times m-reduction [i] would yield (56, 104, 107913)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 169246 664405 989159 886487 607361 149860 978037 433127 250174 335248 551955 594402 796512 955658 439789 726437 812108 329225 088785 057704 541656 > 16¹⁰⁴ [i]