MinT - Best Known (55−21, 55, s)-Nets in Base 16

Best Known (55−21, 55, s)-Nets in Base 16

(55−21, 55, 552)-Net over F₁₆ — Constructive and digital

Digital (34, 55, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (21, 42, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 21, 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]

(55−21, 55, 1144)-Net over F₁₆ — Digital

Digital (34, 55, 1144)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−21, 55, 959689)-Net in Base 16 — Upper bound on s

There is no (34, 55, 959690)-net in base 16, because

1 times m-reduction [i] would yield (34, 54, 959690)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 105313 157327 419475 055711 244152 848510 005695 649939 686626 724467 565376 > 16⁵⁴ [i]