MinT - Best Known (69−25, 69, s)-Nets in Base 16

Best Known (69−25, 69, s)-Nets in Base 16

(69−25, 69, 579)-Net over F₁₆ — Constructive and digital

Digital (44, 69, 579)-net over F₁₆, using

16¹ times duplication [i] based on digital (43, 68, 579)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 18, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (25, 50, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(69−25, 69, 1905)-Net over F₁₆ — Digital

Digital (44, 69, 1905)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−25, 69, 2347553)-Net in Base 16 — Upper bound on s

There is no (44, 69, 2347554)-net in base 16, because

1 times m-reduction [i] would yield (44, 68, 2347554)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7588 557378 634349 645172 525064 297576 539645 391873 121484 673989 685413 589262 246063 965096 > 16⁶⁸ [i]