MinT - Best Known (68−31, 68, s)-Nets in Base 16

Best Known (68−31, 68, s)-Nets in Base 16

(68−31, 68, 520)-Net over F₁₆ — Constructive and digital

Digital (37, 68, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 34, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(68−31, 68, 642)-Net over F₁₆ — Digital

Digital (37, 68, 642)-net over F₁₆, using

2 times m-reduction [i] based on digital (37, 70, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 35, 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]

(68−31, 68, 102339)-Net in Base 16 — Upper bound on s

There is no (37, 68, 102340)-net in base 16, because

1 times m-reduction [i] would yield (37, 67, 102340)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 307149 568276 856772 405394 582432 950916 974898 554817 231839 541058 584654 695599 283376 > 16⁶⁷ [i]