MinT - Best Known (82−25, 82, s)-Nets in Base 16

Best Known (82−25, 82, s)-Nets in Base 16

(82−25, 82, 1045)-Net over F₁₆ — Constructive and digital

Digital (57, 82, 1045)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 8, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (12, 24, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 12, 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 (25, 50, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(82−25, 82, 8511)-Net over F₁₆ — Digital

Digital (57, 82, 8511)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−25, 82, large)-Net in Base 16 — Upper bound on s

There is no (57, 82, large)-net in base 16, because

23 times m-reduction [i] would yield (57, 59, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]