Effective distribution of codewords for Low Density Parity Check Cycle codes in the presence of disorder

We review the zeta-function representation of codewords allowed by a parity-check code based on a bipartite graph, and then investigate the effect of disorder on the effective distribution of codewords. The randomness (or disorder) is implemented by sampling the graph from an ensemble of random graphs, and computing the average zeta function of the ensemble. In the limit of arbitrarily large size for the vertex set of the graph, we find an exponential decay of the likelihood for nontrivial codewords corresponding to graph cycles. This result provides a quantitative estimate of the effect of randomization in cybersecurity applications.