MinT - Best Known (90−41, 90, s)-Nets in Base 16

Best Known (90−41, 90, s)-Nets in Base 16

(90−41, 90, 522)-Net over F₁₆ — Constructive and digital

Digital (49, 90, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 45, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(90−41, 90, 642)-Net over F₁₆ — Digital

Digital (49, 90, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (49, 94, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 47, 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]

(90−41, 90, 126331)-Net in Base 16 — Upper bound on s

There is no (49, 90, 126332)-net in base 16, because

1 times m-reduction [i] would yield (49, 89, 126332)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146795 515643 546174 636916 408678 993580 266932 296995 161347 402837 589079 228269 316861 827851 510403 988702 047477 502226 > 16⁸⁹ [i]