MinT - Best Known (54−21, 54, s)-Nets in Base 16

Best Known (54−21, 54, s)-Nets in Base 16

(54−21, 54, 547)-Net over F₁₆ — Constructive and digital

Digital (33, 54, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(54−21, 54, 997)-Net over F₁₆ — Digital

Digital (33, 54, 997)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−21, 54, 727307)-Net in Base 16 — Upper bound on s

There is no (33, 54, 727308)-net in base 16, because

1 times m-reduction [i] would yield (33, 53, 727308)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6582 093618 608073 858652 514382 067864 257561 834673 509405 176484 729451 > 16⁵³ [i]