Doubly-Exponential Identification via Channels: Code Constructions and Bounds

TL;DR

This paper explores the identification problem over channels, introducing new bounds and code constructions that enable transmitting identifiers with doubly-exponential size, significantly advancing the capacity and efficiency of ID codes.

Contribution

It presents novel bounds on binary constant-weight codes and proposes new code constructions based on optical orthogonal codes and Reed-Solomon codes for improved ID performance.

Findings

New upper bounds on binary CWC size for ID codes.

Proposed code constructions outperform existing codes in certain regimes.

Enhanced finite-parameter performance with larger minimum distance codes.

Abstract

Consider the identification (ID) via channels problem, where a receiver wants to decide whether the transmitted identifier is its identifier, rather than decoding the identifier. This model allows to transmit identifiers whose size scales doubly-exponentially in the blocklength, unlike common transmission (or channel) codes whose size scales exponentially. It suffices to use binary constant-weight codes (CWCs) to achieve the ID capacity. By relating the parameters of a binary CWC to the minimum distance of a code and using higher-order correlation moments, two upper bounds on the binary CWC size are proposed. These bounds are shown to be upper bounds also on the identifier sizes for ID codes constructed by using binary CWCs. We propose two code constructions based on optical orthogonal codes, which are used in optical multiple access schemes, have constant-weight codewords, and satisfy cyclic cross-correlation and auto-correlation constraints. These constructions are modified and concatenated with outer Reed-Solomon codes to propose new binary CWCs optimal for ID. Improvements to the finite-parameter performance of both our and existing code constructions are shown by using outer codes with larger minimum distance vs. blocklength ratios. We also illustrate ID performance regimes for which our ID code constructions perform significantly better than existing constructions.