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Minimum-distance bounds by graph analysis

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1 Author(s)
Tanner, R.M. ; Dept. of Comput. & Inf. Sci., California Univ., Santa Cruz, CA, USA

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]

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Information Theory, IEEE Transactions on  (Volume:47 ,  Issue: 2 )