# 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.

## Key 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.