MinT - Best Known (23, 40, s)-Nets in Base 16

Best Known (23, 40, s)-Nets in Base 16

(23, 40, 520)-Net over F₁₆ — Constructive and digital

Digital (23, 40, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 20, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(23, 40, 643)-Net over F₁₆ — Digital

Digital (23, 40, 643)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16⁴⁰, 643, F₁₆, 2, 17) (dual of [(643, 2), 1246, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(16³⁸, 642, F₁₆, 2, 17) (dual of [(642, 2), 1246, 18]-NRT-code), using
  - extracting embedded OOA [i] based on digital (21, 38, 642)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 19, 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]

(23, 40, 186069)-Net in Base 16 — Upper bound on s

There is no (23, 40, 186070)-net in base 16, because

1 times m-reduction [i] would yield (23, 39, 186070)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 91347 301091 693983 677717 064688 465868 608916 272026 > 16³⁹ [i]