MinT - Best Known (69−27, 69, s)-Nets in Base 16

Best Known (69−27, 69, s)-Nets in Base 16

(69−27, 69, 547)-Net over F₁₆ — Constructive and digital

Digital (42, 69, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 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 (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 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]

(69−27, 69, 1116)-Net over F₁₆ — Digital

Digital (42, 69, 1116)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−27, 69, 751250)-Net in Base 16 — Upper bound on s

There is no (42, 69, 751251)-net in base 16, because

1 times m-reduction [i] would yield (42, 68, 751251)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7588 631705 271218 543831 854193 826039 749837 169051 458030 733795 676175 164436 690864 895496 > 16⁶⁸ [i]