Graph theoretic error-correcting codes

Hakimi, S. L.; Bredeson, J.G.

doi:10.1109/tit.1968.1054190

Cited by 55 publications

(42 citation statements)

References 4 publications

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“…When F is the binary field, the code C[G] is the cut-set code of G, i.e., the dual of the cycle code of G [10]. The following fundamental result that relates the treewidths of the graph G and the code C[G] is due to Hliněný and Whittle 6 [16].…”

Section: Relating the Complexity Measures For Codes And Graphsmentioning

confidence: 99%

On Minimal Tree Realizations of Linear Codes

Kashyap

2009

IEEE Trans. Inform. Theory

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ABSTRACT. A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations.It is known that among all tree realizations of a code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.

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Section: Relating the Complexity Measures For Codes And Graphsmentioning

confidence: 99%

On Minimal Tree Realizations of Linear Codes

Kashyap

2009

IEEE Trans. Inform. Theory

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“…Parity-check matrix based encoding of binary cycle codes Even though it is well-known that binary cycle codes are linearly encodable [1], here we provide a novel proof for this important fact. The encoding method described in the proof will be extended to the encoding of nonbinary cycle codes in Section IV-B.…”

Section: Parallel Sparse Encodable Nonbinary Ldpc Cycle Codesmentioning

confidence: 84%

“…It is well known that a cycle code can be represented by normal graphs [1], [7], where each row of the parity-check matrix H corresponds to a vertex and each column corresponds to an edge whose two end vertices correspond to the two rows with nonzero elements in that column. This motivates us to use cage graphs to construct mother matrices with large girths.…”

Section: A Construction Of the Mother Matrixmentioning

confidence: 99%

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Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes

Peng

Chen

2007

2007 IEEE Information Theory Workshop

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Abstract-In this paper, we study the design of nonbinary low-density parity-check (LDPC) cycle codes over Galois field GF(q). First, we construct a special class of nonbinary LDPC cycle codes with low error floors. Our construction utilizes the cycle elimination algorithm to remove short cycles in the normal graph and to select nonzero elements in the parity-check matrix to reduce the number of low-weight codewords generated by short cycles. Furthermore, we show that simple modifications of such codes are parallel sparse encodable (PSE). The PSE code, consisting of a quasi-cyclic (QC) LDPC cycle code and a simple tree code, has the attractive feature that it is not only linearly encodable, but also allows parallel encoding which can reduce the encoding time significantly. We provide a systematic comparison between nonbinary coded systems and binary coded systems. For the MIMO channel considered, our results show that the proposed nonbinary system employing the PSE code outperforms not only the binary LDPC code specified in the 802.16e standard, but also the optimized binary LDPC code obtained using the EXIT chart methods.

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“…Let q be a prime power and consider the representation of the cycle matroid M (Γ) over the field F q by the columns of a matrix G. The cocycle code C(Γ) [9] is the row space of G. The length of C(Γ) equals n; the dimension is |V | − c(E).…”

Section: ) Is the Minimum Dimension Of A Binary Linear Code Of Lengthmentioning

confidence: 99%

On Some Polynomials Related to Weight Enumerators of Linear Codes

Barg¹

2002

SIAM J. Discrete Math.

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Abstract.A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model. Key words. all-terminal reliability, Greene's theorem, linear code, linear matroid, Potts model, rank polynomial, Tutte polynomial, weight enumerator AMS subject classifications. 94B05, 05B35PII. S0895480199364148 Introduction.A linear matroid M together with a chosen representation over a finite field F q is the same object as a linear code. The most well-known result underlining this connection is Greene's theorem on the relation of the weight polynomial of the code and the Tutte polynomial of the matroid. In this paper we further examine the relation between the polynomial invariants of codes, matroids, and some other combinatorial objects. Our point of view is coding-theoretic. We begin with listing basic definitions for linear codes and some very simple linear-algebraic properties of subcodes. These properties lead almost immediately to a relation between the weight polynomial of a linear code and the rank polynomial of the corresponding matroid. This relation is equivalent to Greene's theorem which is shown to be a purely linear-algebraic fact. An advantage of the coding-theoretic point of view is determined by the fact that the weight polynomial enjoys more structural properties than more general matroid invariants; when this structure translates to other problems, it sometimes produces interesting insights.As an example, we relate the reliability polynomial of a linear matroid to an evaluation of the weight polynomial of the code. The corresponding functional on linear codes turns out to be well studied under the name of the probability of undetected error of the code. Together with some related ideas this enables us to derive upper and lower bounds on the matroid reliability. As another application of the weightrank connection, we give a direct proof of the link between the partition function of the Potts model and the weight polynomial of the cocycle code of the graph.General sources for coding theory are the books [17], [19]. Relevant applications of the Tutte polynomial are covered in [6], [24]. All the necessary information on interaction models is contained in [24].

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Graph theoretic error-correcting codes

Cited by 55 publications

References 4 publications

On Minimal Tree Realizations of Linear Codes

On Minimal Tree Realizations of Linear Codes

Design of Nonbinary Quasi-Cyclic LDPC Cycle Codes

On Some Polynomials Related to Weight Enumerators of Linear Codes

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