MinT - Best Known (72−27, 72, s)-Nets in Base 16

Best Known (72−27, 72, s)-Nets in Base 16

(72−27, 72, 563)-Net over F₁₆ — Constructive and digital

Digital (45, 72, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 18, 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 (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 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]

(72−27, 72, 1532)-Net over F₁₆ — Digital

Digital (45, 72, 1532)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−27, 72, 1424493)-Net in Base 16 — Upper bound on s

There is no (45, 72, 1424494)-net in base 16, because

1 times m-reduction [i] would yield (45, 71, 1424494)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 082889 371267 044923 699406 928530 402036 539476 794197 612658 700400 109049 232380 019842 103331 > 16⁷¹ [i]