MinT - Best Known (89, 89+37, s)-Nets in Base 16

Best Known (89, 89+37, s)-Nets in Base 16

(89, 89+37, 1073)-Net over F₁₆ — Constructive and digital

Digital (89, 126, 1073)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 16, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (18, 36, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 18, 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 (37, 74, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 37, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(89, 89+37, 15616)-Net over F₁₆ — Digital

Digital (89, 126, 15616)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+37, large)-Net in Base 16 — Upper bound on s

There is no (89, 126, large)-net in base 16, because

35 times m-reduction [i] would yield (89, 91, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]