# Error Exponent Bounds for the Bee-Identification Problem

## TL;DR

This paper introduces the bee-identification problem, defines its error exponent, and derives bounds showing joint decoding outperforms separate decoding, with tight bounds at low rates.

## Contribution

It formally defines the bee-identification problem, introduces error exponent bounds, and demonstrates the superiority of joint decoding over separate decoding.

## Key findings

- Joint decoding significantly improves error exponents.
- Lower bounds using typical random codes outperform random code ensembles.
- Bounds converge at zero rate, indicating optimality of joint decoding in that regime.

## Abstract

Consider the problem of identifying a massive number of bees, uniquely labeled with barcodes, using noisy measurements. We formally introduce this `bee-identification problem', define its error exponent, and derive efficiently computable upper and lower bounds for this exponent. We show that joint decoding of barcodes provides a significantly better exponent compared to separate decoding followed by permutation inference. For low rates, we prove that the lower bound on the bee-identification exponent obtained using typical random codes (TRC) is strictly better than the corresponding bound obtained using a random code ensemble (RCE). Further, as the rate approaches zero, we prove that the upper bound on the bee-identification exponent meets the lower bound obtained using TRC with joint barcode decoding.