MinT - Best Known (54, 84, s)-Nets in Base 16

Best Known (54, 84, s)-Nets in Base 16

(54, 84, 581)-Net over F₁₆ — Constructive and digital

Digital (54, 84, 581)-net over F₁₆, using

1 times m-reduction [i] based on 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, 84, 594)-Net in Base 16 — Constructive

(54, 84, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (9, 24, 80)-net in base 16, using
  - base change [i] based on digital (1, 16, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 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]

(54, 84, 2407)-Net over F₁₆ — Digital

Digital (54, 84, 2407)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 84, 2369986)-Net in Base 16 — Upper bound on s

There is no (54, 84, 2369987)-net in base 16, because

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]