MinT - Best Known (4, 51, s)-Nets in Base 16

Best Known (4, 51, s)-Nets in Base 16

(4, 51, 45)-Net over F₁₆ — Constructive and digital

Digital (4, 51, 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]

(4, 51, 187)-Net in Base 16 — Upper bound on s

There is no (4, 51, 188)-net in base 16, because

1 times m-reduction [i] would yield (4, 50, 188)-net in base 16, but
- extracting embedded orthogonal array [i] would yield OA(16⁵⁰, 188, S₁₆, 46), but
  - the linear programming bound shows that MÂ â‰¥ 1091 765195 970395 890315 693716 185672 889915 998980 326190 139213 332068 390649 880226 998373 056512 / 657 761924 179275 666558 951967 > 16⁵⁰ [i]