MinT - Best Known (57−42, 57, s)-Nets in Base 4

Best Known (57−42, 57, s)-Nets in Base 4

(57−42, 57, 33)-Net over F₄ — Constructive and digital

Digital (15, 57, 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]

(57−42, 57, 35)-Net over F₄ — Digital

Digital (15, 57, 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]

(57−42, 57, 74)-Net in Base 4 — Upper bound on s

There is no (15, 57, 75)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁷, 75, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 175541 891167 451460 543215 010075 896682 965259 780096 / 6 746458 181029 > 4⁵⁷ [i]