MinT - Best Known (57, 57+34, s)-Nets in Base 16

Best Known (57, 57+34, s)-Nets in Base 16

(57, 57+34, 579)-Net over F₁₆ — Constructive and digital

Digital (57, 91, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 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 (34, 68, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 34, 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]

(57, 57+34, 1852)-Net over F₁₆ — Digital

Digital (57, 91, 1852)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+34, 1334855)-Net in Base 16 — Upper bound on s

There is no (57, 91, 1334856)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 37 576688 894320 864388 841645 032102 671950 665274 176641 461043 194295 322529 111003 815848 972749 406460 968093 709569 087756 > 16⁹¹ [i]