2006/08/05 by Alexandros G. Dimakis, Martin J. Wainwright, Dimakis, Alexandros G. +1 · 2 citations
Computer Science · Engineering · #Advanced Wireless Communication Techniques #Coding theory and cryptography #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.cs/0608029
openalex publication_date 2006/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A new approach for decoding binary linear codes by solving a linear program (LP) over a relaxed codeword polytope was recently proposed by Feldman et al. In this paper we investigate the structure of the polytope used in the LP relaxation decoding. We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword. We exploit this fact to devise a decoding algorithm that provably outperforms the LP decoder for finite blocklengths. It proceeds by guessing facets of the polytope, and resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that for expander codes the proposed algorithm succeeds even with a constant number of pseudocodewords of higher likelihood. Moreover, the complexity of the proposed algorithm is only a constant factor larger than that of the LP decoder.