MinT - Best Known (127−50, 127, s)-Nets in Base 16

Best Known (127−50, 127, s)-Nets in Base 16

(127−50, 127, 547)-Net over F₁₆ — Constructive and digital

Digital (77, 127, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 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 (50, 100, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 50, 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]

(127−50, 127, 1708)-Net over F₁₆ — Digital

Digital (77, 127, 1708)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−50, 127, 888090)-Net in Base 16 — Upper bound on s

There is no (77, 127, 888091)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 837 991474 141782 039627 704562 840034 918332 430464 393963 204867 567267 257693 917997 733708 525148 234264 608707 958806 142423 262486 371162 262103 565928 922641 081895 539126 > 16¹²⁷ [i]