MinT - Best Known (111−53, 111, s)-Nets in Base 16

Best Known (111−53, 111, s)-Nets in Base 16

(111−53, 111, 518)-Net over F₁₆ — Constructive and digital

Digital (58, 111, 518)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 112, 518)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 56, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(111−53, 111, 642)-Net over F₁₆ — Digital

Digital (58, 111, 642)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 112, 642)-net over F₁₆, using
- trace code for nets [i] based on digital (2, 56, 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]

(111−53, 111, 87394)-Net in Base 16 — Upper bound on s

There is no (58, 111, 87395)-net in base 16, because

1 times m-reduction [i] would yield (58, 110, 87395)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 839512 078160 917678 222182 737371 867997 758818 799904 323113 571334 699683 290589 991444 768810 407920 912547 037381 429996 321242 368919 291906 174676 > 16¹¹⁰ [i]