Trade-offs between Error Exponents and Excess-Rate Exponents of Typical Slepian–Wolf Codes

Abstract

Typical random codes (TRCs) in a communication scenario of source coding with side information in the decoder is the main subject of this work. We study the semi-deterministic code ensemble, which is a certain variant of the ordinary random binning code ensemble. In this code ensemble, the relatively small type classes of the source are deterministically partitioned into the available bins in a one-to-one manner. As a consequence, the error probability decreases dramatically. The random binning error exponent and the error exponent of the TRCs are derived and proved to be equal to one another in a few important special cases. We show that the performance under optimal decoding can be attained also by certain universal decoders, e.g., the stochastic likelihood decoder with an empirical entropy metric. Moreover, we discuss the trade-offs between the error exponent and the excess-rate exponent for the typical random semi-deterministic code and characterize its optimal rate function. We show that for any pair of correlated information sources, both error and excess-rate probabilities exponential vanish when the blocklength tends to infinity.

Reversing Jensen’s Inequality for Information-Theoretic Analyses

“…A Gallager-style exponent was developed in [186]. The results were then extended to the colored Gaussian channel in [17], to random time-varying trellis codes in [187], and to typical SW codes in [188]. In [189], the TCEM was used to establish that a stochastic MMI decoder, which is a universal decoder, achieves the exponent of the typical random code and the expurgated exponent.…”