Spectral graph analysis of quasi-cyclic codes

Authors:
Roxana Smarandache;Mark F. Flanagan
Affiliations:
Department of Mathematics and Statistics, San Diego State University, San Diego, CA;Department of Electronic Engineering, University College Dublin, Belfield, Dublin 4, Ireland
Venue:
GLOBECOM'09 Proceedings of the 28th IEEE conference on Global telecommunications
Year:
2009

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Abstract

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 λ1 λ2 of the eigenvalues of HTH: wPmin (H) ≥ n 2c-λ2/λ1-λ2. [6]. 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 HTH 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.