Block Network Error Control Codes and Syndrome-based Complete Maximum Likelihood Decoding

In this paper, network error control coding is studied for robust and efficient multicast in a directed acyclic network with imperfect links. The block network error control coding framework, BNEC, is presented and the capability of the scheme to correct a mixture of symbol errors and packet erasures and to detect symbol errors is studied. The idea of syndrome-based decoding and error detection is introduced for BNEC, which removes the effect of input data and hence decreases the complexity. Next, an efficient three-stage syndrome-based BNEC decoding scheme for network error correction is proposed, in which prior to finding the error values, the position of the edge errors are identified based on the error spaces at the receivers. In addition to bounded-distance decoding schemes for error correction up to the refined Singleton bound, a complete decoding scheme for BNEC is also introduced. Specifically, it is shown that using the proposed syndrome-based complete decoding, a network error correcting code with redundancy order δ t for receiver t, can correct δ t-1 random additive errors with a probability sufficiently close to 1, if the field size is sufficiently large. Also, a complete maximum likelihood decoding scheme for BNEC is proposed. As the probability of error in different network edges is not equal in general, and given the equivalency of certain edge errors within the network at a particular receiver, the number of edge errors, assessed in the refined Singleton bound, is not a sufficient statistic for ML decoding. I. INTRODUCTION The joint design of network and channel codes is an enabling technique for robust and efficient communications over networks with imperfect links. In this paradigm, this article deals with the design of block network error control codes and syndrome-based error correction and detection in a directed acyclic network. More specifically, we consider the scenario in which the network is subject to a mixture of additive random symbol errors and packet erasures with possibly different probabilities in different links.