On Network-Error Correcting Convolutional Codes under the BSC Edge Error Model

TL;DR

This paper analyzes the performance of convolutional network-error correcting codes over networks with edges modeled as independent binary symmetric channels, providing bounds, conditions, and code design insights for different error regimes.

Contribution

It introduces bounds and conditions for CNECC performance under BSC edge errors, and identifies code properties needed for low and high error probabilities.

Findings

Convolutional codes require different properties for low and high error regimes.

Derived bounds on code performance based on transfer function analysis.

Established a lower bound on the slope of rate b/c convolutional codes.

Abstract

Convolutional network-error correcting codes (CNECCs) are known to provide error correcting capability in acyclic instantaneous networks within the network coding paradigm under small field size conditions. In this work, we investigate the performance of CNECCs under the error model of the network where the edges are assumed to be statistically independent binary symmetric channels, each with the same probability of error $p_e$($0\leq p_e<0.5$). We obtain bounds on the performance of such CNECCs based on a modified generating function (the transfer function) of the CNECCs. For a given network, we derive a mathematical condition on how small $p_e$ should be so that only single edge network-errors need to be accounted for, thus reducing the complexity of evaluating the probability of error of any CNECC. Simulations indicate that convolutional codes are required to possess different properties to achieve good performance in low $p_e$ and high $p_e$ regimes. For the low $p_e$ regime, convolutional codes with good distance properties show good performance. For the high $p_e$ regime, convolutional codes that have a good \textit{slope} (the minimum normalized cycle weight) are seen to be good. We derive a lower bound on the slope of any rate $b/c$ convolutional code with a certain degree.