Error Exponents of the Dirty-Paper and Gel'fand-Pinsker Channels

TL;DR

This paper derives and compares various error exponents for the Gel'fand-Pinsker and dirty-paper channels, revealing improvements at low rates and differences in optimal parameters for capacity achievement.

Contribution

It provides new error exponent analyses for these channels, including sub-optimal decoders and variable-rate coding, enhancing understanding of their performance limits.

Findings

Error exponents are improved at low coding rates.

Sub-optimal decoders are asymptotically optimal for error exponents.

Optimal design parameters differ between error exponents and capacity.

Abstract

We derive various error exponents for communication channels with random states, which are available non-causally at the encoder only. For both the finite-alphabet Gel'fand-Pinsker channel and its Gaussian counterpart, the dirty-paper channel, we derive random coding exponents, error exponents of the typical random codes (TRCs), and error exponents of expurgated codes. For the two channel models, we analyze some sub-optimal bin-index decoders, which turn out to be asymptotically optimal, at least for the random coding error exponent. For the dirty-paper channel, we show explicitly via a numerical example, that both the error exponent of the TRC and the expurgated exponent strictly improve upon the random coding exponent, at relatively low coding rates, which is a known fact for discrete memoryless channels without random states. We also show that at rates below capacity, the optimal values of the dirty-paper design parameter $\alpha$ in the random coding sense and in the TRC exponent sense are different from one another, and they are both different from the optimal $\alpha$ that is required for attaining the channel capacity. For the Gel'fand-Pinsker channel, we allow for a variable-rate random binning code construction, and prove that the previously proposed maximum penalized mutual information decoder is asymptotically optimal within a given class of decoders, at least for the random coding error exponent.