  Kazhdan and Lusztig define, for an arbitrary Coxeter system $(W,S)$, a family
of polynomials indexed by pairs of elements of $W$. Despite their relevance and
elementary definition, the explicit computation of these polynomials is still
one of the hardest open problems in algebraic combinatorics. In this paper we
explicitly compute Kazhdan-Lusztig polynomials for a Coxeter system of type
$\tilde{B}_2$.
