MinT - Best Known (42, 42+28, s)-Nets in Base 16

Best Known (42, 42+28, s)-Nets in Base 16

(42, 42+28, 531)-Net over F₁₆ — Constructive and digital

Digital (42, 70, 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 (28, 56, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 28, 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]

(42, 42+28, 978)-Net over F₁₆ — Digital

Digital (42, 70, 978)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+28, 422628)-Net in Base 16 — Upper bound on s

There is no (42, 70, 422629)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 942702 643951 949919 895686 226385 394592 437867 444522 138147 936367 757713 151376 903017 811216 > 16⁷⁰ [i]