MinT - Best Known (109−31, 109, s)-Nets in Base 16

Best Known (109−31, 109, s)-Nets in Base 16

(109−31, 109, 1093)-Net over F₁₆ — Constructive and digital

Digital (78, 109, 1093)-net over F₁₆, using

16¹ times duplication [i] based on digital (77, 108, 1093)-net over F₁₆, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 16, 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]
  2. digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 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 (31, 62, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(109−31, 109, 19055)-Net over F₁₆ — Digital

Digital (78, 109, 19055)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−31, 109, large)-Net in Base 16 — Upper bound on s

There is no (78, 109, large)-net in base 16, because

29 times m-reduction [i] would yield (78, 80, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]