Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

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 survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes

In this paper, we present an improved union bound on the Linear Programming (LP) decoding performance of the binary linear codes transmitted over an additive white Gaussian noise channels. The bounding technique is based on the second-order of Bonferroni-type inequality in probability theory, and it is minimized by Prim's minimum spanning tree algorithm. The bound calculation needs the fundamental cone generators of a given parity-check matrix rather than only their weight spectrum, but involves relatively low computational complexity. It is targeted to high-density parity-check codes, where the number of their generators is extremely large and these generators are spread densely in the Euclidean space. We explore the generator density and make a comparison between different parity-check matrix representations. That density effects on the improvement of the proposed bound over the conventional LP union bound. The paper also presents a complete pseudo-weight distribution of the fundamental cone generators for the BCH[31,21,5] code

Distance Properties of Short LDPC Codes and their Impact on the BP, ML and Near-ML Decoding Performance

Parameters of LDPC codes, such as minimum distance, stopping distance, stopping redundancy, girth of the Tanner graph, and their influence on the frame error rate performance of the BP, ML and near-ML decoding over a BEC and an AWGN channel are studied. Both random and structured LDPC codes are considered. In particular, the BP decoding is applied to the code parity-check matrices with an increasing number of redundant rows, and the convergence of the performance to that of the ML decoding is analyzed. A comparison of the simulated BP, ML, and near-ML performance with the improved theoretical bounds on the error probability based on the exact weight spectrum coefficients and the exact stopping size spectrum coefficients is presented. It is observed that decoding performance very close to the ML decoding performance can be achieved with a relatively small number of redundant rows for some codes, for both the BEC and the AWGN channels

Improved linear programming decoding of LDPC codes and bounds on the minimum and fractional distance

We examine LDPC codes decoded using linear programming (LP). Four contributions to the LP framework are presented. First, a new method of tightening the LP relaxation, and thus improving the LP decoder, is proposed. Second, we present an algorithm which calculates a lower bound on the minimum distance of a specific code. This algorithm exhibits complexity which scales quadratically with the block length. Third, we propose a method to obtain a tight lower bound on the fractional distance, also with quadratic complexity, and thus less than previously-existing methods. Finally, we show how the fundamental LP polytope for generalized LDPC codes and nonbinary LDPC codes can be obtained.Comment: 17 pages, 8 figures, Submitted to IEEE Transactions on Information Theor

Lower bounds on the minimum average distance of binary codes

New lower bounds on the minimum average Hamming distance of binary codes are derived. The bounds are obtained using linear programming approach.Comment: 19 page

On the lengths of divisible codes

In this article, the effective lengths of all $q^r$-divisible linear codes over $\mathbb{F}_q$ with a non-negative integer $r$ are determined. For that purpose, the $S_q(r)$-adic expansion of an integer $n$ is introduced. It is shown that there exists a $q^r$-divisible $\mathbb{F}_q$-linear code of effective length $n$ if and only if the leading coefficient of the $S_q(r)$-adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^r$-divisible code of effective length $n$ is at most $\sigma q^r$, where $\sigma$ denotes the cross-sum of the $S_q(r)$-adic expansion of $n$. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace codes