Minimum-distance bounds by graph analysis

Abstract

The parity-check matrix of a linear code is used to define a bipartite code constraint (Tanner) graph in which bit nodes are connected to parity check nodes. The connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix. A simple lower bound on minimum distance of the code is expressed in terms of the two largest eigenvalues. For a more powerful bound, local properties of the subgraph corresponding to a minimum weight word in the code are used to create an optimization problem whose solution is a lower bound on the code's minimum distance. Linear programming gives one bound. The technique is illustrated by applying it to sparse block codes with parameters [7,3,4] and [42,23,6].

Combinatorial constructions of low-density parity check codes for iterative decoding

“…Our bounds are derived using simple combinatorial arguments, and are solely a function of the parity-check matrix column weight. Tanner's bounds on the minimum distance based on the eigenvalues of the product of the adjacency matrix of the code graph and its transpose [39] can be shown to be trivial for the codes presented in this paper.…”

“…Bounds for of Gallager codes with column weight were first derived in [38]. Another, more general technique for establishing a lower bound for is due to Tanner [39]. It pertains to an arbitrary linear code with parity-check matrix , represented by a bipartite graph, and is based on combinatorial optimization.…”

“…Based on Lemma 3.1, one can use Tanner's lower bounds [39] to find bounds on the minimum distance of the CDF codes. Tanner's bounds are expressed in terms of the second largest eigenvalue of (see also [61]), and for the case of interest are of the form The first bound is trivial for , i.e., for .…”

Small weight codewords in the LDPC codes arising from linear representations of geometries

“…On the other hand, the code arising from the dual geometry T * 2 (K) D of T * 2 (K) is a (ρ = s + 1,γ = t + 1)-LDPC code of length q 3 . In Table 1, we have denoted by Θ, B, U and L a hyperoval, a Baer subplane, a Hermitian curve, and two intersecting lines, respectively, and we have presented the lower bounds on the minimum distance d min of the LDPC codes arising from their linear representations due to the bit-oriented bound, the parity oriented bound and Massey's bound [26]. The main goal of this section is to find the minimum distance of these LDPC codes and to characterize their small weight codewords.…”

The Minimum Distance of Graph Codes

“…In 1981 Tanner [1] introduced a construction of error-correcting codes based on graphs and since then a considerable number of results have been obtained [2], [3], [4], [5] and [6]. The recent textbook by Roth [8] contains a thorough presentation of the subject.…”