MinT - Best Known (89−56, 89, s)-Nets in Base 3

Best Known (89−56, 89, s)-Nets in Base 3

(89−56, 89, 38)-Net over F₃ — Constructive and digital

Digital (33, 89, 38)-net over F₃, using

t-expansion [i] based on digital (32, 89, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(89−56, 89, 46)-Net over F₃ — Digital

Digital (33, 89, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(89−56, 89, 120)-Net in Base 3 — Upper bound on s

There is no (33, 89, 121)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁹, 121, S₃, 56), but
- the linear programming bound shows that MÂ â‰¥ 558 813775 281789 332098 447012 287925 760277 888019 085660 572057 993929 / 162 087228 760122 712865 > 3⁸⁹ [i]