MinT - Best Known (75, 75+43, s)-Nets in Base 16

Best Known (75, 75+43, s)-Nets in Base 16

(75, 75+43, 583)-Net over F₁₆ — Constructive and digital

Digital (75, 118, 583)-net over F₁₆, using

16¹ times duplication [i] based on digital (74, 117, 583)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 27, 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 (47, 90, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 45, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(75, 75+43, 2680)-Net over F₁₆ — Digital

Digital (75, 118, 2680)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+43, 2958476)-Net in Base 16 — Upper bound on s

There is no (75, 118, 2958477)-net in base 16, because

1 times m-reduction [i] would yield (75, 117, 2958477)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 762 148878 569930 272121 104961 340616 734194 259295 122932 034139 601331 103117 890668 114700 692592 960263 095153 076525 278251 884627 647602 207780 905496 648256 > 16¹¹⁷ [i]