MinT - Best Known (82−32, 82, s)-Nets in Base 16

Best Known (82−32, 82, s)-Nets in Base 16

(82−32, 82, 547)-Net over F₁₆ — Constructive and digital

Digital (50, 82, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 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 (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]

(82−32, 82, 1283)-Net over F₁₆ — Digital

Digital (50, 82, 1283)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−32, 82, 672290)-Net in Base 16 — Upper bound on s

There is no (50, 82, 672291)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 546 821840 203763 444480 566926 463089 372488 571789 255955 677400 749405 366077 640925 057746 116002 178965 414966 > 16⁸² [i]