MinT - Best Known (27, 27+74, s)-Nets in Base 16

Best Known (27, 27+74, s)-Nets in Base 16

(27, 27+74, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 101, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 101, 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]

(27, 27+74, 76)-Net in Base 16 — Constructive

(27, 101, 76)-net in base 16, using

9 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
- base change [i] based on digital (5, 88, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(27, 27+74, 156)-Net over F₁₆ — Digital

Digital (27, 101, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(27, 27+74, 1870)-Net in Base 16 — Upper bound on s

There is no (27, 101, 1871)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 469045 856486 224122 871445 858978 041872 106298 421401 199433 435687 223631 853285 977326 402541 432983 052147 188777 295894 973187 648756 > 16¹⁰¹ [i]