  In this paper we study the hardness of the syndrome decoding problem over
finite rings endowed with the Lee metric. We first prove that the decisional
version of the problem is NP-complete, by a reduction from the $3$-dimensional
matching problem. Then, we study the complexity of solving the problem, by
translating the best known solvers in the Hamming metric over finite fields to
the Lee metric over finite rings, as well as proposing some novel solutions.
For the analyzed algorithms, we assess the computational complexity in the
asymptotic regime and compare it to the corresponding algorithms in the Hamming
metric.
