Error Exponents for Variable-length Block Codes with Feedback and Cost Constraints

Variable-length block-coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints. Upper and lower bounds are found for the minimum achievable probability of decoding error \(P_{e,\min}\) as a function of constraints \(R, \AV\), and \(\bar \tau\) on the transmission rate, average cost, and average block length respectively. For given \(R\) and \(\AV\), the lower and upper bounds to the exponent \(-(\ln P_{e,\min})/\bar \tau\) are asymptotically equal as \(\bar \tau \to \infty\). The resulting reliability function, \(\lim_{\bar \tau\to \infty} (-\ln P_{e,\min})/\bar \tau\), as a function of \(R\) and \(\AV\), is concave in the pair \((R, \AV)\) and generalizes the linear reliability function of Burnashev to include cost constraints. The results are generalized to a class of discrete-time memoryless channels with arbitrary alphabets, including additive Gaussian noise channels with amplitude and power constraints.