MinT - Best Known (37, 37+22, s)-Nets in Base 16

Best Known (37, 37+22, s)-Nets in Base 16

(37, 37+22, 559)-Net over F₁₆ — Constructive and digital

Digital (37, 59, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (22, 44, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 22, 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]

(37, 37+22, 1408)-Net over F₁₆ — Digital

Digital (37, 59, 1408)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+22, 940550)-Net in Base 16 — Upper bound on s

There is no (37, 59, 940551)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110428 989149 667091 397425 117120 552326 705668 384387 234095 811531 490801 116416 > 16⁵⁹ [i]