MinT - Best Known (64−24, 64, s)-Nets in Base 16

Best Known (64−24, 64, s)-Nets in Base 16

(64−24, 64, 559)-Net over F₁₆ — Constructive and digital

Digital (40, 64, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (24, 48, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 24, 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]

(64−24, 64, 1421)-Net over F₁₆ — Digital

Digital (40, 64, 1421)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−24, 64, 931623)-Net in Base 16 — Upper bound on s

There is no (40, 64, 931624)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115792 759431 911396 254919 476840 951041 942371 641087 880933 656643 719694 055944 439196 > 16⁶⁴ [i]