MinT - Best Known (65, 65+57, s)-Nets in Base 16

Best Known (65, 65+57, s)-Nets in Base 16

(65, 65+57, 522)-Net over F₁₆ — Constructive and digital

Digital (65, 122, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 61, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(65, 65+57, 642)-Net over F₁₆ — Digital

Digital (65, 122, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (65, 126, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 63, 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]

(65, 65+57, 120330)-Net in Base 16 — Upper bound on s

There is no (65, 122, 120331)-net in base 16, because

1 times m-reduction [i] would yield (65, 121, 120331)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 49 950278 086344 692331 978676 154402 963407 111737 894571 179735 830012 914015 949981 173113 820185 614067 910404 813836 144054 590977 336897 233787 705315 082760 269696 > 16¹²¹ [i]