MinT - Best Known (54, 54+31, s)-Nets in Base 16

Best Known (54, 54+31, s)-Nets in Base 16

(54, 54+31, 581)-Net over F₁₆ — Constructive and digital

Digital (54, 85, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (33, 64, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 32, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(54, 54+31, 2087)-Net over F₁₆ — Digital

Digital (54, 85, 2087)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+31, 2369986)-Net in Base 16 — Upper bound on s

There is no (54, 85, 2369987)-net in base 16, because

1 times m-reduction [i] would yield (54, 84, 2369987)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 139984 748488 003872 415405 256154 331877 054808 477081 776304 569065 075014 807300 027691 583482 109099 227602 688576 > 16⁸⁴ [i]