## Mathematics, Department of

## Date of this Version

2008

## Citation

xvig, N., Dreher, D., Morrison, K., Psota, E., Perez, L. C., & Walker, J. L. (2008). Towards universal cover decoding. In *2008 International Symposium on Information Theory and its Applications, ISITA 2008 *

doi 10.1109/ISITA.2008.4895386

## Abstract

Low complexity decoding of low-density paritycheck (LDPC) codes may be obtained from the application of iterative message-passing decoding algorithms to the bipartite Tanner graph of the code. Arguably, the two most important decoding algorithms for LDPC codes are the sum-product decoder and the min-sum (MS) decoder. On a bipartite graph without cycles (a tree), the sum-product decoder minimizes the probability of bit error, while the min-sum decoder minimizes the probability of word error [9]. While the behavior of sum-product and min-sum is easily understood when operating on trees, their behavior becomes much more difficult to characterize when the Tanner graph has cycles. Wiberg [9] showed that decoding can be modeled by finding minimal cost configurations on computation trees that are formed at successive iterations of sum-product/min-sum, and returning the value assigned to the root nodes of these trees. Additionally, he proved that for an error to occur at a particular variable node, there must exist a deviation of non-positive cost on the computation tree rooted at this node. In this paper, we are interested in analyzing the non-codeword errors that occur during parallel, iterative decoding with the min-sum decoder. Recently, work has been done relating the min-sum decoder to the linear programming (LP) decoder via graph covers [8]. The LP decoder, as defined by Feldman [3], recasts the problem of decoding as an optimization problem whose feasible set is a polytope defined by the parity-check matrix of a code. In [8], it is shown that LP decoding can be realized as a decoder operating on graph covers. The notion that non-codeword outputs of LP decoding are related to non-codeword outputs of min-sum decoding is attractive from an analytical perspective. However, the performance of LP and min-sum are not consistently related [2]. Therefore, a different theoretical model is needed to explore the relationship between decoding on graph covers and decoding on computation trees. To bridge this gap, we will turn to the notion of decoding on the universal cover. Universal covers can be thought of as both infinite computation trees and infinite graph covers. For this reason, decoding on universal covers provides an intuitive link between LP decoding and min-sum decoding of LDPC codes. This paper is an extension of previous work done by the authors in [2]; thus, much of the requisite background material is drawn from [2]. Section 2 introduces the definition of universal covers. Properties related to configurations on universal covers and their corresponding costs are established in Section 3. Finally, in Section 4 a preliminary definition of the universal cover decoder is given, and it is shown that under certain conditions the universal cover decoder agrees with the LP decoder.