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, P, and τ on the transmission rate, average cost, and average block length respectively. For given R and P, the lower and upper bounds to the exponent −(ln P e,min )/τ are asymptotically equal as τ → ∞. The resulting reliability function, lim τ →∞ (− ln P e,min )/τ , as a function of R and P, is concave in the pair (R, P) and generalizes the linear reliability function of Burnashev [2] 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.
Nakiboglu and Gallager REF investigated variable-length coding schemes for (not necessarily discrete) memoryless channels with variable-length feedback and with cost constraints and established the reliability function.
Error Exponents for Variable-length Block Codes with Feedback and Cost Constraints