MinT - Best Known (81, 81+33, s)-Nets in Base 16

Best Known (81, 81+33, s)-Nets in Base 16

(81, 81+33, 1077)-Net over F₁₆ — Constructive and digital

Digital (81, 114, 1077)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 16, 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 (33, 66, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 33, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(81, 81+33, 16630)-Net over F₁₆ — Digital

Digital (81, 114, 16630)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+33, large)-Net in Base 16 — Upper bound on s

There is no (81, 114, large)-net in base 16, because

31 times m-reduction [i] would yield (81, 83, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]