Probabilistic Analysis of Linear Programming Decoding

Abstract—We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of “probabilistic expansion” in random bipartite graphs, in which one requires only that almost...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Constantinos Daskalakis, Alexandros G. Dimakis, Richard M. Karp, Martin J. Wainwright
• Keywords: Analysis, CORR 2007, Education, LDPC Codes, Linear Programming