MinT - Best Known (54−39, 54, s)-Nets in Base 4

Best Known (54−39, 54, s)-Nets in Base 4

(54−39, 54, 33)-Net over F₄ — Constructive and digital

Digital (15, 54, 33)-net over F₄, using

net from sequence [i] based on digital (15, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
  - an explicitly constructive algebraic function field [i]

(54−39, 54, 35)-Net over F₄ — Digital

Digital (15, 54, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(54−39, 54, 87)-Net in Base 4 — Upper bound on s

There is no (15, 54, 88)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁴, 88, S₄, 39), but
- the linear programming bound shows that MÂ â‰¥ 388 306151 532706 712327 958115 613049 308256 005391 927015 697876 516864 / 1 038662 238351 660386 161900 190625 > 4⁵⁴ [i]