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

Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes

Linear programming (LP) decoding for low-density parity-check codes was introduced by Feldman et al. and has been shown to have theoretical guarantees in several regimes. Furthermore, it has been reported in the literature—via simulation and via instanton analysis—that LP decoding displays better error rate performance at high signal-to-noise ratios (SNR) than does belief propagation (BP) decoding. However, at low SNRs, LP decoding is observed to have worse performance than BP. In this paper, we seek to improve LP decoding at low SNRs while maintaining LP decoding’s high SNR performance. Our main contribution is a new class of decoders obtained by applying the alternating direction method of multipliers (ADMM) algorithm to a set of non-convex optimization problems. These non-convex problems are constructed by adding a penalty term to the objective of LP decoding. The goal of the penalty is to make pseudocodewords, which are non-integer vertices of the LP relaxation, more costly. We name this class of decoders—ADMM penalized decoders. For low and moderate SNRs, we simulate ADMM penalized decoding with $\ell _{1}$ and $\ell _{2}$ penalties. We find that these decoders can outperform both BP and LP decoding. For high SNRs, where it is difficult to obtain data via simulation, we use an instanton analysis and find that, asymptotically, ADMM penalized decoding performs better than BP but not as well as LP. Unfortunately, since ADMM penalized decoding is not a convex program, we have not been successful in developing theoretical guarantees. However, the non-convex program can be approximated using a sequence of linear programs; an approach that yields a reweighted LP decoder. We show that a two-round reweighted LP decoder has an improved theoretical recovery threshold when compared with LP decoding. In addition, we find via simulation that reweighted LP decoding significantly attains lower error rates than LP decoding at low SNRs.

The ADMM Penalized Decoder for LDPC Codes

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Linear programming decoding with the alternating direction method of multipliers (ADMM) is a promising decoding technique for low-density parity-check (LDPC) codes, where the computational complexity of Euclidean projections onto check polytopes becomes a prominent problem. In this paper, the problem is circumvented by building lookup tables (LUTs) and quantizing the inputs to approach approximate Euclidean projections at low computational complexities. To challenge the huge memory cost of LUTs, we first propose two commutative compositions of Euclidean projection and self-map, and show the existence of a small quantization range which does not alter the Euclidean projection. Then, we investigate the design and simplification of the LUTs by exploiting the commutative compositions and check node decomposition techniques. An efficient algorithm for the LUT-based projection is demonstrated by using one simplification method. Simulation results show that for both the regular and irregular LDPC codes, the ADMM decoding using LUT-based projection can substantially reduce the decoding time while maintaining the error rate performance at a comparatively large memory cost.

Efficient ADMM Decoding of LDPC Codes Using Lookup Tables

In this paper we present a three-stage decoding strategy that combines quantized and un-quantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerfulMILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added. Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated “active-set” version of the quantized BP decoder we significantly speed up the pace of simulation to simulate low density parity check (LDPC) codes of length up to around 2000 down to a WER of around 10−10 on the binary symmetric channel. We demonstrate that for certain codes our approach can efficiently approach the optimal ML decoding performance for low crossover probabilities.

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Multi-stage decoding of LDPC codes

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.

A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

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.

In recent years several authors have used Integer Programming (IP) formulations to model the Maximum Likelihood (ML) decoding problem. IP formulation of the ML decoding problem can be modified to calculate the minimum distance of the code also. The IP based approach for MD calculation can be effectively used for small block length codes. However, it is not known if this approach is useful for the moderate and long length codes used in practice. In this paper we demonstrate the usefulness of the IP based approach for the MD calculation by applying it to practically relevant codes. We calculate either the true MD or upper bounds on the MD for 3GPP LTE turbo codes, WiMax duo-binary turbo codes and WiMax & WiFi low density parity-check codes. A new IP formulation is also introduced to calculate the MD of punctured turbo codes. Along with MD calculation, the same IP formulation is used to find multiplicity of low weight codewords. Our results prove that, the IP based approach can estimate the MD of practically relevant codes fairly quickly and can also provide information about the multiplicity of codewords with a given Hamming weight. Also, the IP based method for the MD calculation is independent of the code used and can easily accommodate puncturing schemes.

Calculating the minimum distance of linear block codes via Integer Programming

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.

Akin Tanatmis

Papers

A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Calculating the minimum distance of linear block codes via Integer Programming

Valid inequalities for binary linear codes