MinT - Best Known (68−30, 68, s)-Nets in Base 16

Best Known (68−30, 68, s)-Nets in Base 16

(68−30, 68, 522)-Net over F₁₆ — Constructive and digital

Digital (38, 68, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 34, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(68−30, 68, 642)-Net over F₁₆ — Digital

Digital (38, 68, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (38, 72, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 36, 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]

(68−30, 68, 123118)-Net in Base 16 — Upper bound on s

There is no (38, 68, 123119)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7588 723637 101830 868999 201611 422017 490460 857511 234714 300926 576789 383731 028545 221776 > 16⁶⁸ [i]