Minimum-distance bounds by graph analysis

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].