MinT - Best Known (90−27, 90, s)-Nets in Base 16

Best Known (90−27, 90, s)-Nets in Base 16

(90−27, 90, 1052)-Net over F₁₆ — Constructive and digital

Digital (63, 90, 1052)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 10, 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 (13, 26, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 13, 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]
3. 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 (see above)

(90−27, 90, 10372)-Net over F₁₆ — Digital

Digital (63, 90, 10372)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−27, 90, large)-Net in Base 16 — Upper bound on s

There is no (63, 90, large)-net in base 16, because

25 times m-reduction [i] would yield (63, 65, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]