On the Characterization of Linear Uniquely Decodable Codes

A Uniquely Decodable (UD) Code is a code such that any vector of the ambient space has a unique closest codeword. In this paper we begin a study of the structure of UD codes and identify perfect subcodes. In particular we determine all linear UD codes of covering radius 2. 1 Introduction We call n-cube or hypercube of dimension n the graph whose vertices are all the binary vectors of length n, two vertices being adjacent if their Hamming distance is one. If a subgraph \Gamma of the n-cube IF n is isomorphic to the r- subcube IF r we will say that \Gamma is an r-subcube (r n). Clearly an r- subcube of IF n is obtained by taking all vectors that have some fixed values in n \Gamma r fixed coordinate positions. A Perfect Dominating Set (PDS) S of a graph \Gamma is a set of vertices of \Gamma such that every vertex of \Gamma is either in S or adjacent to exactly one vertex of S. Removing "exactly" yields a Dominating Set (DS). The latter appears in quite a variety of contexts..