Spectral Graph Analysis of Quasi-Cyclic Codes

TL;DR

This paper investigates spectral properties of quasi-cyclic codes to establish bounds on their pseudo-weight over AWGNC, providing methods to compute eigenvalues efficiently and identifying conditions for optimal bounds.

Contribution

It introduces a spectral analysis approach for quasi-cyclic codes, reducing eigenvalue computation complexity and deriving conditions for achieving bounds on pseudo-weight.

Findings

Eigenvalues of large matrices can be reduced to smaller matrices.

A necessary condition for the bound to be attained is identified.

Certain classes of cyclic codes satisfy the optimality criterion.

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 eigenvalues of H^T H. In particular, we analyze (c,d)-regular quasi-cyclic (QC) codes of length rL described by J x L block parity-check matrices with circulant block entries of size r x r. We proceed by showing how the problem of computing the eigenvalues of the rL x rL matrix H^T H can be reduced to the problem of computing eigenvalues for r matrices of size L x 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.