MinT - Best Known (44, 44+39, s)-Nets in Base 16

Best Known (44, 44+39, s)-Nets in Base 16

(44, 44+39, 518)-Net over F₁₆ — Constructive and digital

Digital (44, 83, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 84, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 42, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(44, 44+39, 642)-Net over F₁₆ — Digital

Digital (44, 83, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (44, 84, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 42, 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]

(44, 44+39, 83137)-Net in Base 16 — Upper bound on s

There is no (44, 83, 83138)-net in base 16, because

1 times m-reduction [i] would yield (44, 82, 83138)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 827154 292585 267061 867353 211330 621588 416374 097418 775077 773843 819015 218940 728357 440961 856855 313056 > 16⁸² [i]