MinT - Best Known (51, 51+45, s)-Nets in Base 16

Best Known (51, 51+45, s)-Nets in Base 16

(51, 51+45, 520)-Net over F₁₆ — Constructive and digital

Digital (51, 96, 520)-net over F₁₆, using

trace code for nets [i] based on digital (3, 48, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(51, 51+45, 642)-Net over F₁₆ — Digital

Digital (51, 96, 642)-net over F₁₆, using

2 times m-reduction [i] based on digital (51, 98, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 49, 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]

(51, 51+45, 95568)-Net in Base 16 — Upper bound on s

There is no (51, 96, 95569)-net in base 16, because

1 times m-reduction [i] would yield (51, 95, 95569)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 462704 336908 969745 795164 286925 044611 808234 198715 463083 488869 140610 660496 022055 937887 496324 717837 418136 773091 044896 > 16⁹⁵ [i]