MinT - Best Known (92−57, 92, s)-Nets in Base 16

Best Known (92−57, 92, s)-Nets in Base 16

(92−57, 92, 89)-Net over F₁₆ — Constructive and digital

Digital (35, 92, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 29, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 63, 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]

(92−57, 92, 120)-Net in Base 16 — Constructive

(35, 92, 120)-net in base 16, using

28 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
- base change [i] based on digital (11, 96, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(92−57, 92, 193)-Net over F₁₆ — Digital

Digital (35, 92, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 92, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(92−57, 92, 6154)-Net in Base 16 — Upper bound on s

There is no (35, 92, 6155)-net in base 16, because

1 times m-reduction [i] would yield (35, 91, 6155)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 37 592625 827612 888513 103890 100634 120526 996321 629760 984184 989618 453118 641406 015818 773177 230181 154098 934315 838976 > 16⁹¹ [i]