List decoding of Convolutional Codes over integer residue rings

Abstract

where Z p r [D] stands for the ring of polynomials with coefficients in Z p r . In this paper we study the list decoding problem of these codes when the transmission is performed over an erasure channel, that is, we study how much information one can recover from a codeword w ∈ C when some of its coefficients have been erased. We do that using the p-adic expansion of w and particular representations of the parity-check matrix of the code. From these matrix representations we recursively select certain equations that w must satisfy and have only coefficients in the field p r−1 Z p r . This yields a step by step procedure to obtain a list of possible codewords for a given corrupted codeword w. We show that such an algorithm actually computes all possible erased coordinates, that is, it provides a minimal list with the closest codewords to the vector w. Mathematically, this problem amounts to determine the set of all possible solutions of a set of linear equations over Z p r [D] that can be represented by a matrix with Toeplitz structure.