MinT - Best Known (77, 77+47, s)-Nets in Base 16

Best Known (77, 77+47, s)-Nets in Base 16

(77, 77+47, 579)-Net over F₁₆ — Constructive and digital

Digital (77, 124, 579)-net over F₁₆, using

16¹ times duplication [i] based on digital (76, 123, 579)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 29, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
  2. digital (47, 94, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 47, 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]

(77, 77+47, 2137)-Net over F₁₆ — Digital

Digital (77, 124, 2137)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+47, 1728964)-Net in Base 16 — Upper bound on s

There is no (77, 124, 1728965)-net in base 16, because

1 times m-reduction [i] would yield (77, 123, 1728965)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12786 732500 639743 041809 589863 034160 873981 492292 286689 541691 824063 264443 365093 855431 293026 897907 112552 297220 637047 508543 552097 132080 693027 093608 730176 > 16¹²³ [i]