Deterministic identification for Bernoulli channels and related channels with continuous input

TL;DR

This paper extends recent code constructions to determine the exact deterministic identification capacity for a broad class of channels, including Bernoulli and channels with continuous output distributions, closing longstanding capacity gaps.

Contribution

It proves the tightness of the converse bound for these channels and establishes the deterministic identification capacity as 1/2, advancing understanding of DI for continuous-input channels.

Findings

Established the DI capacity as 1/2 for a broad class of channels.

Closed the long-standing gap between achievability and converse bounds.

Derived improved lower bounds on the reliability function.

Abstract

For memoryless channels with continuous input alphabets, deterministic identification (DI) typically exhibits a linearithmic ($n\log n$) message growth. However, the exact DI capacity has long remained open due to a persistent gap between the best known achievability and converse bounds. This gap was recently closed for AWGN channels via a novel code construction optimising the "galaxy" codes. Here, we extend this approach to the Bernoulli channel and subsequently to any channel $W$ whose image contains a continuous curve of output probability distributions, and hence admits a reduction to the Bernoulli channel restricted to a subinterval of inputs. As a consequence, we prove that the converse bound is tight and establish $\dot{C}_{\text{DI}}(W) = \frac 12$ for this broad class of channels, thereby closing the long-standing capacity gap. A similar gap was also observed for the DI rate-reliability tradeoff. We analyse the tradeoff between rate and error of the proposed code and derive improved lower bounds on the reliability function, approaching the converse at leading order in the regime of small error exponents.