MinT - Best Known (45, 45+78, s)-Nets in Base 3

Best Known (45, 45+78, s)-Nets in Base 3

(45, 45+78, 48)-Net over F₃ — Constructive and digital

Digital (45, 123, 48)-net over F₃, using

net from sequence [i] based on digital (45, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(45, 45+78, 56)-Net over F₃ — Digital

Digital (45, 123, 56)-net over F₃, using

t-expansion [i] based on digital (40, 123, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(45, 45+78, 147)-Net in Base 3 — Upper bound on s

There is no (45, 123, 148)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹²³, 148, S₃, 78), but
- the linear programming bound shows that MÂ â‰¥ 7 262897 028362 273678 472870 511072 492435 452301 893929 213302 547823 936352 413873 / 144 349303 060703 > 3¹²³ [i]