MinT - Best Known (52, 52+45, s)-Nets in Base 16

Best Known (52, 52+45, s)-Nets in Base 16

(52, 52+45, 520)-Net over F₁₆ — Constructive and digital

Digital (52, 97, 520)-net over F₁₆, using

1 times m-reduction [i] based on digital (52, 98, 520)-net over F₁₆, using
- trace code for nets [i] based on digital (3, 49, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(52, 52+45, 642)-Net over F₁₆ — Digital

Digital (52, 97, 642)-net over F₁₆, using

3 times m-reduction [i] based on digital (52, 100, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 50, 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]

(52, 52+45, 108406)-Net in Base 16 — Upper bound on s

There is no (52, 97, 108407)-net in base 16, because

1 times m-reduction [i] would yield (52, 96, 108407)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 39 405503 338398 232972 870995 695386 309838 877091 598001 555233 723899 034198 166091 639927 685530 876521 562106 943194 429817 194611 > 16⁹⁶ [i]