MinT - Best Known (112−69, 112, s)-Nets in Base 16

Best Known (112−69, 112, s)-Nets in Base 16

(112−69, 112, 225)-Net over F₁₆ — Constructive and digital

Digital (43, 112, 225)-net over F₁₆, using

t-expansion [i] based on digital (40, 112, 225)-net over F₁₆, using
- net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
    - a function field by GarcÃa and Quoos [i]

(112−69, 112, 226)-Net over F₁₆ — Digital

Digital (43, 112, 226)-net over F₁₆, using

net from sequence [i] based on digital (43, 225)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 43 and N(F) ≥ 226, using
  - Drinfeld modules of rank 1 [i]

(112−69, 112, 7681)-Net in Base 16 — Upper bound on s

There is no (43, 112, 7682)-net in base 16, because

1 times m-reduction [i] would yield (43, 111, 7682)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 611157 608198 925058 197513 013572 228104 767836 842695 035341 185296 631601 653906 922855 347226 759702 392747 878580 108174 435110 816088 335501 450496 > 16¹¹¹ [i]