MinT - Best Known (126−61, 126, s)-Nets in Base 16

Best Known (126−61, 126, s)-Nets in Base 16

(126−61, 126, 518)-Net over F₁₆ — Constructive and digital

Digital (65, 126, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 63, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(126−61, 126, 642)-Net over F₁₆ — Digital

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]

(126−61, 126, 83517)-Net in Base 16 — Upper bound on s

There is no (65, 126, 83518)-net in base 16, because

1 times m-reduction [i] would yield (65, 125, 83518)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 273813 180851 625322 250727 714156 524694 387227 936716 672404 665534 856962 652464 592478 715807 471558 448682 243241 747004 043999 108293 022728 539898 961937 392338 300976 > 16¹²⁵ [i]