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 the 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].