MinT - Best Known (56−25, 56, s)-Nets in Base 16

Best Known (56−25, 56, s)-Nets in Base 16

(56−25, 56, 520)-Net over F₁₆ — Constructive and digital

Digital (31, 56, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 28, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(56−25, 56, 642)-Net over F₁₆ — Digital

Digital (31, 56, 642)-net over F₁₆, using

2 times m-reduction [i] based on digital (31, 58, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 29, 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]

(56−25, 56, 116447)-Net in Base 16 — Upper bound on s

There is no (31, 56, 116448)-net in base 16, because

1 times m-reduction [i] would yield (31, 55, 116448)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 685095 392113 094146 060718 077566 564617 324349 945296 444009 716654 926741 > 16⁵⁵ [i]