MinT - Best Known (61−23, 61, s)-Nets in Base 16

Best Known (61−23, 61, s)-Nets in Base 16

(61−23, 61, 559)-Net over F₁₆ — Constructive and digital

Digital (38, 61, 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 (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]

(61−23, 61, 1328)-Net over F₁₆ — Digital

Digital (38, 61, 1328)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(61−23, 61, 1210174)-Net in Base 16 — Upper bound on s

There is no (38, 61, 1210175)-net in base 16, because

1 times m-reduction [i] would yield (38, 60, 1210175)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 766854 189969 765937 891799 409159 626239 222768 953574 352438 648349 230862 378876 > 16⁶⁰ [i]