MinT - Best Known (54, 54+32, s)-Nets in Base 16

Best Known (54, 54+32, s)-Nets in Base 16

(54, 54+32, 579)-Net over F₁₆ — Constructive and digital

Digital (54, 86, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 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 (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 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, 54+32, 1828)-Net over F₁₆ — Digital

Digital (54, 86, 1828)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+32, 1344589)-Net in Base 16 — Upper bound on s

There is no (54, 86, 1344590)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 836274 492032 126825 041947 198651 235627 522373 265994 858996 719809 110526 703207 163990 739815 525171 955314 113476 > 16⁸⁶ [i]