On the Hardness of the Lee Syndrome Decoding Problem

We consider codes over finite rings endowed with the Lee metric and prove the NP-completeness of the associated syndrome decoding problem (SDP), by reduction from the shortest path problem in circulant graphs. With analogous arguments and via randomized reduction, we also prove the hardness of deciding whether a given code contains codewords with bounded Lee weight. Then, we study the best known algorithms for solving the SDP, which are information set decoding (ISD) algorithms, and generalize them to the Lee metric case. Finally we assess their complexity for a wide range of parameters. Our results suggest that, for an arbitrary code, decoding up to the error correction capability given by the Gilbert-Varshamov bound in the Lee metric is much more difficult than in its Hamming metric counterpart.