MinT - Best Known (76, 76+47, s)-Nets in Base 16

Best Known (76, 76+47, s)-Nets in Base 16

(76, 76+47, 579)-Net over F₁₆ — Constructive and digital

Digital (76, 123, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 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 (47, 94, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 47, 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]

(76, 76+47, 2013)-Net over F₁₆ — Digital

Digital (76, 123, 2013)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(76, 76+47, 1532613)-Net in Base 16 — Upper bound on s

There is no (76, 123, 1532614)-net in base 16, because

1 times m-reduction [i] would yield (76, 122, 1532614)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 799 172163 257239 402385 069170 326463 015445 431643 046213 609398 688004 793578 922581 357378 434357 310927 913157 641801 679910 127954 967906 611123 161656 444951 391456 > 16¹²² [i]