MinT - Best Known (57, 110, s)-Nets in Base 16

Best Known (57, 110, s)-Nets in Base 16

(57, 110, 518)-Net over F₁₆ — Constructive and digital

Digital (57, 110, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 55, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(57, 110, 642)-Net over F₁₆ — Digital

Digital (57, 110, 642)-net over F₁₆, using

trace code for nets [i] based on digital (2, 55, 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]

(57, 110, 78553)-Net in Base 16 — Upper bound on s

There is no (57, 110, 78554)-net in base 16, because

1 times m-reduction [i] would yield (57, 109, 78554)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 177490 575687 322161 060420 263417 138222 958331 185190 619705 801521 894762 009748 432939 159876 120749 722146 663983 444337 640441 002650 838265 007936 > 16¹⁰⁹ [i]