MinT - Best Known (56, 56+37, s)-Nets in Base 16

Best Known (56, 56+37, s)-Nets in Base 16

(56, 56+37, 538)-Net over F₁₆ — Constructive and digital

Digital (56, 93, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (37, 74, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 37, 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]

(56, 56+37, 1246)-Net over F₁₆ — Digital

Digital (56, 93, 1246)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+37, 718495)-Net in Base 16 — Upper bound on s

There is no (56, 93, 718496)-net in base 16, because

1 times m-reduction [i] would yield (56, 92, 718496)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 601 236811 203712 706782 412932 018015 827012 704656 326009 276296 494333 203083 919884 834559 428002 985731 921326 733523 546971 > 16⁹² [i]