MMSE of "Bad" Codes

TL;DR

This paper investigates 'bad' codes over Gaussian channels constrained by MMSE, revealing that superposition codebooks achieve maximum rate and characterizing MMSE and mutual information behavior across SNRs.

Contribution

It demonstrates that superposition codebooks maximize rate under MMSE constraints and characterizes MMSE and mutual information for all SNRs for such codes.

Findings

Superposition codebooks attain the maximum rate under MMSE constraints.

The MMSE and mutual information behavior is fully characterized for all SNRs.

A lower bound on MMSE for finite-length codes based on error probability is provided.

Abstract

We examine codes, over the additive Gaussian noise channel, designed for reliable communication at some specific signal-to-noise ratio (SNR) and constrained by the permitted minimum mean-square error (MMSE) at lower SNRs. The maximum possible rate is below point-to-point capacity, and hence these are non-optimal codes (alternatively referred to as "bad" codes). We show that the maximum possible rate is the one attained by superposition codebooks. Moreover, the MMSE and mutual information behavior as a function of SNR, for any code attaining the maximum rate under the MMSE constraint, is known for all SNR. We also provide a lower bound on the MMSE for finite length codes, as a function of the error probability of the code.