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International Symposium on Turbo Codes and Iterative Information Processing

A separation algorithm for improved LP-decoding of linear block codes

Akın Tanatmis, Stefan Ruzika, Horst W. Hamacher, Mayur Punekar, Frank Kienle, Norbert Wehn

Published Sep 1, 2008• 5 citations
Publisher

Abstract

We present improvements in linear programming decoding introduced by Feldman et al. with the help of generic integer programming methods. We propose a new IP formulation of the maximum likelihood decoding problem. This formulation makes use of indicator variables which help in detecting violated parity checks immediately. We derive Gomory cuts from our formulation and use them in a separation algorithm to find maximum likelihood codewords. We further propose a novel method of finding cuts induced by redundant parity checks. Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. We demonstrate the performance of our separation algorithm on two LDPC codes and one BCH code.

DOI
10.1109/turbocoding.2008.4658669

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IEEE Transactions on Information Theory•Feb 29, 2008•78 citations
We present improvements in linear programming decoding introduced by Feldman et al. with the help of generic integer programming methods. We propose a new IP formulation of the maximum likelihood decoding problem. This formulation makes use of indicator variables which help in detecting violated parity checks immediately. We derive Gomory cuts from our formulation and use them in a separation algorithm to find maximum likelihood codewords. We further propose a novel method of finding cuts induced by redundant parity checks. Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. We demonstrate the performance of our separation algorithm on two LDPC codes and one BCH code.
Maximum Likelihood (ML) decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an Integer Programming (IP) problem. Feldman et al. relaxed this IP problem and presented Linear Programming (LP) based decoding algorithm for linear block codes. In this paper, we propose a new IP formulation of the ML decoding problem and solve the IP with generic methods. The formulation uses indicator variables to detect violated parity checks. We derive Gomory cuts from our formulation and use them in a separation algorithm to find ML codewords. We further propose an efficient method of finding cuts induced by redundant parity checks (RPC). Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. We demonstrate on two LDPC codes and one BCH code that our separation algorithm performs significantly better than LP decoding.
Maximum likelihood (ML) decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming (IP) problem. Feldman relaxed this IP problem and presented linear programming (LP) based decoding. In this paper, we propose a new separation algorithm to improve the error-correcting performance of LP decoding for binary linear block codes. We use an IP formulation with indicator variables that help in detecting the violated parity checks. We derive Gomory cuts from the IP and use them in our separation algorithm. An efficient method of finding cuts induced by redundant parity checks (RPC) is also proposed. Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. It is demonstrated on three LDPC codes and two BCH codes that our separation algorithm performs significantly better than LP decoding and belief propagation (BP) decoding.
Abstract—Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities ” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut, ” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding. Index Terms—Iterative decoding, linear codes, linear programming (LP) decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, pseudocodewords. I.
We study an integer programming (IP) based separation approach to find the maximum likelihood (ML) codeword for binary linear codes. An algorithm introduced in Tanatmis et al. is extended and improved with respect to decoding performance without increasing the worst case complexity. This is demonstrated on the LDPC and the BCH code classes. Moreover, we propose an integer programming formulation to calculate the minimum distance of a binary linear code. We exemplarily compute the minimum distance of the (204, 102) LDPC code and the (576, 288) WIMAX code. Using the minimum distance of a code, a new class of valid inequalities is introduced.
Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called “parity inequalities” that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or “cut,” at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding.
In this paper, we propose a new linear programming formulation for the decoding of general linear block codes. Different from the original formulation given by Feldman, the number of total variables to characterize a parity-check constraint in our formulation is less than twice the degree of the corresponding check node. The equivalence between our new formulation and the original formulation is proven. The new formulation facilitates to characterize the structure of linear block codes, and leads to new decoding algorithms. In particular, we show that any fundamental polytope is simply the intersection of a group of the so-called minimum polytopes, and this simplified formulation allows us to formulate the problem of calculating the minimum Hamming distance of any linear block code as a simple linear integer programming problem with much less auxiliary variables. We then propose a branch-and-bound method to compute a lower bound to the minimum distance of any linear code by solving a corresponding linear integer programming problem. In addition, we prove that, for the family of single parity-check (SPC) product codes, the fractional distance and the pseudodistance are both equal to the minimum distance. Finally, we propose an efficient algorithm for decoding SPC product codes with low complexity and maximum-likelihood (ML) decoding performance.