Random-coding error exponent of variable-length codes with a single-bit noiseless feedback

We study the random-coding error exponent function of variable-length codes in the presence of a noiseless feedback channel, which is allowed to be used merely for a single bit feedback per each transmitted message. In this study, we harness results and analysis techniques from the theory of sequential hypothesis testing, and combine them with modern distance enumeration methods which are used in the literature on error exponents. For this setup, sometimes referred to as stop-feedback, we derive an exact single-letter expression for the random-coding error exponent over the binary symmetric channel. For symmetric discrete memoryless channels, the exact error exponent at zero rate is obtained, and a lower bound is provided for any other positive rate below capacity.