A Branch-Price-and-Cut Algorithm for Optimal Decoding of LDPC Codes

Channel coding aims to minimize errors that occur during the transmission of digital information from one place to another. Low-density parity-check (LDPC) codes can detect and correct transmission errors if one encodes the original information by adding redundant bits. In practice, heuristic iterative decoding algorithms are used to decode the received vector. However, these algorithms may fail to decode if the received vector contains multiple errors. We consider decoding the received vector with minimum error as an integer programming problem and propose a branch-and-price method for its solution. We improve the performance of our method by introducing heuristic feasible solutions and adding valid cuts to the mathematical formulation. Computational results reveal that our branch-price-and-cut algorithm significantly improves solvability of the problem compared to a commercial solver in high channel error rates. Our proposed algorithm can find higher quality solutions than commonly used iterative decoding heuristics.Comment: 30 pages, 4 figures, 7 table

Exact separation of forbidden-set cuts associated with redundant parity checks of binary linear codes

In recent years, several integer programming (IP) approaches were developed for maximum-likelihood decoding and minimum distance computation for binary linear codes. Two aspects in particular have been demonstrated to improve the performance of IP solvers as well as adaptive linear programming decoders: the dynamic generation of forbidden-set (FS) inequalities, a family of valid cutting planes, and the utilization of so-called redundant parity-checks (RPCs). However, to date, it had remained unclear how to solve the exact RPC separation problem (i.e., to determine whether or not there exists any violated FS inequality w.r.t. any known or unknown parity-check). In this note, we prove NP-hardness of this problem. Moreover, we formulate an IP model that combines the search for most violated FS cuts with the generation of RPCs, and report on computational experiments. Empirically, for various instances of the minimum distance problem, it turns out that while utilizing the exact separation IP does not appear to provide a computational advantage, it can apparently be avoided altogether by combining heuristics to generate RPC-based cuts