Gallager bounds for linear codes in binary-input output-symmetric memoryless channels

This paper presents a general methodology to extend Gallager bounds on the maximum-likelihood decoding error probability to arbitrary binary-input output-symmetric memoryless channels. Based on the log-likelihood ratios, a new space is constructed in which the signals naturally lie on a sphere, and for which geometric analysis is straightforward. In particular, we focus on Poltyrev's tangential-sphere bound, and we illustrate its connections with the Engdahl-Zigangirov bound. Approximations to these bounds are shown to be very tight.