MinT - Best Known (98−45, 98, s)-Nets in Base 16

Best Known (98−45, 98, s)-Nets in Base 16

(98−45, 98, 522)-Net over F₁₆ — Constructive and digital

Digital (53, 98, 522)-net over F₁₆, using

trace code for nets [i] based on digital (4, 49, 261)-net over F₂₅₆, using
- net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(98−45, 98, 642)-Net over F₁₆ — Digital

Digital (53, 98, 642)-net over F₁₆, using

4 times m-reduction [i] based on digital (53, 102, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 51, 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]

(98−45, 98, 122968)-Net in Base 16 — Upper bound on s

There is no (53, 98, 122969)-net in base 16, because

1 times m-reduction [i] would yield (53, 97, 122969)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 630 485258 358571 352248 533177 121936 505727 605398 248269 086725 579967 495647 968475 352788 687412 364401 837016 496209 097768 079396 > 16⁹⁷ [i]