MinT - Best Known (72−29, 72, s)-Nets in Base 16

Best Known (72−29, 72, s)-Nets in Base 16

(72−29, 72, 531)-Net over F₁₆ — Constructive and digital

Digital (43, 72, 531)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(72−29, 72, 954)-Net over F₁₆ — Digital

Digital (43, 72, 954)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−29, 72, 515191)-Net in Base 16 — Upper bound on s

There is no (43, 72, 515192)-net in base 16, because

1 times m-reduction [i] would yield (43, 71, 515192)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 083115 900566 140379 258097 901592 344761 890126 212648 924989 570100 224062 378873 684312 620746 > 16⁷¹ [i]