Spectral Graph Analysis of Quasi-Cyclic Codes

In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c, d)-regular linear block code based on the two largest values ¿₁ > ¿₂ of the eigenvalues of H^TH: w_p ^min > (H) ¿ n = 2c-¿₂/¿₁-¿₂. In particular, we analyze (c, d)-regular quasi-cyclic (QC) codes of length rL described by J × L block parity-check matrices with circulant block entries of size r × r. We proceed by showing how the problem of computing the eigenvalues of the rL × rL matrix H^TH can be reduced to the problem of computing eigenvalues for r matrices of size L × L. We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion.