The Linear Reliability Channel

We introduce and analyze a discrete soft-decision channel called the linear reliability channel (LRC) in which the soft information is the rank-ordering of the received symbol reliabilities. We prove that the LRC is an appropriate approximation to a general class of binary-input, continuous-output channels when the noise variance is high. The central feature of the LRC is that its combinatorial nature allows for an extensive mathematical analysis of the channel and its corresponding hard- and soft-decision maximum-likelihood (ML) decoders. In particular, we establish explicit error exponents for ML decoding in the LRC when using random codes under both hard- and soft-decision decoding. This analysis allows for a direct, quantitative evaluation of the relative advantage of soft-decision decoding. The discrete geometry of the LRC is distinct from that of the BSC, which is characterized by the Hamming weight, offering a new perspective on code construction for soft-decision settings.