Error Exponents in the Bee Identification Problem

TL;DR

This paper analyzes error exponents in the bee identification problem under various decoding rules, deriving bounds and showing the optimality of certain decoders across different coding rates and channel conditions.

Contribution

It introduces new bounds on error exponents for the bee identification problem, including improvements at low and intermediate coding rates, and establishes the optimality of the universal maximum mutual information decoder.

Findings

Derived bounds on error exponents for naive decoding.

Proposed improved bounds using expurgation techniques.

Showed the universal MMI decoder's optimality across scenarios.

Abstract

We derive various error exponents in the bee identification problem under two different decoding rules. Under na\"ive decoding, which decodes each bee independently of the others, we analyze a general discrete memoryless channel and a relatively wide family of stochastic decoders. Upper and lower bounds to the random coding error exponent are derived and proved to be equal at relatively high coding rates. Then, we propose a lower bound on the error exponent of the typical random code, which improves upon the random coding exponent at low coding rates. We also derive a third bound, which is related to expurgated codes, which turns out to be strictly higher than the other bounds, also at relatively low rates. We show that the universal maximum mutual information decoder is optimal with respect to the typical random code and the expurgated code. Moving further, we derive error exponents under optimal decoding, the relatively wide family of symmetric channels, and the maximum likelihood decoder. We first propose a random coding lower bound, and then, an improved bound which stems from an expurgation process. We show numerically that our second bound strictly improves upon the random coding bound at an intermediate range of coding rates, where a bound derived in a previous work no longer holds.