MinT - Best Known (69, 69+45, s)-Nets in Base 16

Best Known (69, 69+45, s)-Nets in Base 16

(69, 69+45, 547)-Net over F₁₆ — Constructive and digital

Digital (69, 114, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 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 (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(69, 69+45, 1538)-Net over F₁₆ — Digital

Digital (69, 114, 1538)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+45, 923749)-Net in Base 16 — Upper bound on s

There is no (69, 114, 923750)-net in base 16, because

1 times m-reduction [i] would yield (69, 113, 923750)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11629 547299 699696 887687 739970 755987 268820 197561 204232 728388 609090 686145 445801 848365 161328 323010 221566 392585 871429 465353 266004 965241 259376 > 16¹¹³ [i]