MinT - Best Known (51, 51+47, s)-Nets in Base 16

Best Known (51, 51+47, s)-Nets in Base 16

(51, 51+47, 518)-Net over F₁₆ — Constructive and digital

Digital (51, 98, 518)-net over F₁₆, using

trace code for nets [i] based on digital (2, 49, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(51, 51+47, 642)-Net over F₁₆ — Digital

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+47, 75255)-Net in Base 16 — Upper bound on s

There is no (51, 98, 75256)-net in base 16, because

1 times m-reduction [i] would yield (51, 97, 75256)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 630 541812 627146 776059 707314 975350 758307 705662 802906 821768 066033 817945 697699 296796 661738 368934 234883 784207 541934 349621 > 16⁹⁷ [i]