Error Rates of Capacity-Achieving Codes Are Convex

TL;DR

This paper proves that the bit error rate of maximum likelihood detectors in AWGN channels is convex at high SNR and noise levels, including for capacity-achieving codes, with a conjecture that all such codes share this property.

Contribution

It establishes convexity of error rates for arbitrary constellations and codes at high SNR, extending the understanding of error behavior in communication systems.

Findings

Error rates are convex functions of SNR and noise power at high SNR.

Capacity-achieving codes satisfy the high SNR convexity condition.

Convexity holds for arbitrary constellations and bit mappings.

Abstract

Motivated by a wide-spread use of convex optimization techniques, convexity properties of bit error rate of the maximum likelihood detector operating in the AWGN channel are studied for arbitrary constellations and bit mappings, which also includes coding under maximum-likelihood decoding. Under this generic setting, the pairwise probability of error and bit error rate are shown to be convex functions of the SNR and noise power in the high SNR/low noise regime with explicitly-determined boundary. Any code, including capacity-achieving ones, whose decision regions include the hardened noise spheres (from the noise sphere hardening argument in the channel coding theorem) satisfies this high SNR requirement and thus has convex error rates in both SNR and noise power. We conjecture that all capacity-achieving codes have convex error rates.