MinT - Best Known (74−29, 74, s)-Nets in Base 16

Best Known (74−29, 74, s)-Nets in Base 16

(74−29, 74, 547)-Net over F₁₆ — Constructive and digital

Digital (45, 74, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 16, 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 (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(74−29, 74, 1160)-Net over F₁₆ — Digital

Digital (45, 74, 1160)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(74−29, 74, 765575)-Net in Base 16 — Upper bound on s

There is no (45, 74, 765576)-net in base 16, because

1 times m-reduction [i] would yield (45, 73, 765576)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7957 280239 776769 926837 687004 733150 040081 566172 597869 157375 054372 538625 495171 279656 909936 > 16⁷³ [i]