Bounds on Maximum Likelihood Decoding Performance for Linear Codes at Low Rates

For a given linear code (ensemble), upper bounds on the error probability under maximum likelihood decoding are investigated at low rates. A class of symmetric memoryless channels suitable for Bhattacharyya-type bounds, which are particularly of importance at low rates, is introduced. Over a symmetric channel, a lower bound on the error exponent is derived for a given linear code. A sufficient condition for achieving the expurgated exponent, which is the best among known error exponents at low rates, within a fixed discrepancy is given. Over a general discrete memoryless channel, the same analysis provides a lower bound on the average error exponent for an ensemble of coset codes generated by a given linear code. The bounding technique is extended to the case of generalized maximum likelihood decoding with erasure and list- decoding options.