MinT - Best Known (111−32, 111, s)-Nets in Base 16

Best Known (111−32, 111, s)-Nets in Base 16

(111−32, 111, 1077)-Net over F₁₆ — Constructive and digital

Digital (79, 111, 1077)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 15, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (16, 32, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 16, 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 (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(111−32, 111, 16977)-Net over F₁₆ — Digital

Digital (79, 111, 16977)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−32, 111, large)-Net in Base 16 — Upper bound on s

There is no (79, 111, large)-net in base 16, because

30 times m-reduction [i] would yield (79, 81, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]