MinT - Best Known (126−59, 126, s)-Nets in Base 16

Best Known (126−59, 126, s)-Nets in Base 16

(126−59, 126, 522)-Net over F₁₆ — Constructive and digital

Digital (67, 126, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 63, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(126−59, 126, 642)-Net over F₁₆ — Digital

Digital (67, 126, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (67, 130, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 65, 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]

(126−59, 126, 120544)-Net in Base 16 — Upper bound on s

There is no (67, 126, 120545)-net in base 16, because

1 times m-reduction [i] would yield (67, 125, 120545)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 274153 822695 020966 761473 546056 886115 670021 277623 087048 385364 792561 941128 939200 237174 255658 921534 236059 903554 855937 265794 815132 593694 630943 965824 180576 > 16¹²⁵ [i]