MinT - Best Known (81−31, 81, s)-Nets in Base 16

Best Known (81−31, 81, s)-Nets in Base 16

(81−31, 81, 559)-Net over F₁₆ — Constructive and digital

Digital (50, 81, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(81−31, 81, 1447)-Net over F₁₆ — Digital

Digital (50, 81, 1447)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−31, 81, 1131476)-Net in Base 16 — Upper bound on s

There is no (50, 81, 1131477)-net in base 16, because

1 times m-reduction [i] would yield (50, 80, 1131477)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 136000 197375 644033 680475 629640 819184 688689 786902 358940 247277 893763 191225 590630 518865 575429 542576 > 16⁸⁰ [i]