MinT - Best Known (68−20, 68, s)-Nets in Base 16

Best Known (68−20, 68, s)-Nets in Base 16

(68−20, 68, 1061)-Net over F₁₆ — Constructive and digital

Digital (48, 68, 1061)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 8, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (10, 20, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 10, 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 (20, 40, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 20, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(68−20, 68, 10789)-Net over F₁₆ — Digital

Digital (48, 68, 10789)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−20, 68, large)-Net in Base 16 — Upper bound on s

There is no (48, 68, large)-net in base 16, because

18 times m-reduction [i] would yield (48, 50, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]