MinT - Best Known (37, 37+23, s)-Nets in Base 16

Best Known (37, 37+23, s)-Nets in Base 16

(37, 37+23, 552)-Net over F₁₆ — Constructive and digital

Digital (37, 60, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (23, 46, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 23, 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+23, 1172)-Net over F₁₆ — Digital

Digital (37, 60, 1172)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+23, 940550)-Net in Base 16 — Upper bound on s

There is no (37, 60, 940551)-net in base 16, because

1 times m-reduction [i] would yield (37, 59, 940551)-net in base 16, but
- 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]