MinT - Best Known (81−32, 81, s)-Nets in Base 16

Best Known (81−32, 81, s)-Nets in Base 16

(81−32, 81, 538)-Net over F₁₆ — Constructive and digital

Digital (49, 81, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 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 (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 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]

(81−32, 81, 1175)-Net over F₁₆ — Digital

Digital (49, 81, 1175)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−32, 81, 565325)-Net in Base 16 — Upper bound on s

There is no (49, 81, 565326)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 176627 207322 578871 966204 810947 777419 830152 905906 483978 403885 826173 039420 247951 905883 764725 642116 > 16⁸¹ [i]