MinT - Best Known (66−29, 66, s)-Nets in Base 16

Best Known (66−29, 66, s)-Nets in Base 16

(66−29, 66, 522)-Net over F₁₆ — Constructive and digital

Digital (37, 66, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 33, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(66−29, 66, 642)-Net over F₁₆ — Digital

Digital (37, 66, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (37, 70, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 35, 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]

(66−29, 66, 157001)-Net in Base 16 — Upper bound on s

There is no (37, 66, 157002)-net in base 16, because

1 times m-reduction [i] would yield (37, 65, 157002)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 852805 015140 922847 609482 213362 714826 283160 464477 447300 674731 606400 116404 952096 > 16⁶⁵ [i]