On the Optimality of the Hamming Metric for Decoding Block Codes over Binary Additive Noise Channels
Communications , Coding Theory
Most of the basic concepts of algebraic coding theory are derived for the memoryless binary symmetric channel. These concepts do not necessarily hold for time-varying channels or for channels with memory. However, errors in real-life channels seem to occur in bursts rather than independently, suggesting that these channels exhibit some statistical dependence or memory. Nonetheless, the same algebraic codes are still commonly used in current communication systems that employ interleaving to spread channel error bursts over the set of received codewords to make the channel appear memoryless to the block decoder. This method suffers from immediate shortcomings as it fails to exploit the channel’s memory while adding delay to the system. We study optimal maximum likelihood block decoding of binary codes sent over several binary additive channels with infinite and finite memory. We derive conditions on general binary codes and channels parameters under which maximum likelihood and minimum distance decoding are equivalent. The channels considered in this work are the infinite and finite memory Polya contagion channels, the queue-based channel, and the Gilbert-Elliott channel. We also present results on the optimality of classical perfect and quasi-perfect codes when used over the aforementioned channels under maximum likelihood decoding.