MinT - Best Known (77, 77+29, s)-Nets in Base 16

Best Known (77, 77+29, s)-Nets in Base 16

(77, 77+29, 1544)-Net over F₁₆ — Constructive and digital

Digital (77, 106, 1544)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 18, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 9, 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]
2. digital (14, 28, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 14, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
3. digital (31, 60, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(77, 77+29, 27264)-Net over F₁₆ — Digital

Digital (77, 106, 27264)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+29, large)-Net in Base 16 — Upper bound on s

There is no (77, 106, large)-net in base 16, because

27 times m-reduction [i] would yield (77, 79, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]