Random Access Codes: Explicit Constructions, Optimality, and Classical-Quantum Gaps

2026-04-23

AI Summary

This work addresses the explicit optimal construction of classical random access codes (RACs) for arbitrary parameters $(L,k)$, with the goal of maximizing both average and worst-case decoding success probabilities, while also examining the performance gap relative to quantum RACs. By reformulating the problem as selecting $2^k$ points in $\{0,1\}^L$ or $[0,1]^L$ to minimize inter-class distances, the authors develop a general framework for explicit constructions. Their main contributions include the first explicit optimal construction for classical RACs in the general $(L,k)$ setting, a rigorous proof that the $(L,L-1)$ case achieves the known theoretical upper bound—and consequently that the corresponding quantum RAC attains the conjectured upper bound—and numerical evidence showing that classical and quantum RACs exhibit comparable performance on average, yet display a significant gap in worst-case scenarios.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A random access code (RAC) encodes an $L$-bit string into a $k$-bit $(L>k)$ message from which any designated source bit can be recovered with high probability. Its quantum counterpart, a quantum random access code (QRAC), replaces the $k$-bit message with $k$ qubits. While upper bounds on the decoding success probability have long been studied in both classical and quantum settings, explicit constructions of optimal codes are known only in special cases, even for classical RACs. In this paper, we develop a constructive framework for classical $(L,k)$-RACs under both average- and worst-case criteria. We show that optimal code design reduces to selecting $2^k$ points in $\{0,1\}^L$ and $[0,1]^L$ for the average- and worst-case criteria, respectively, so as to minimize a distance-like objective. This characterization yields explicit constructions for general $(L,k)$. For $k=L-1$, we further obtain closed-form optimal encoders and decoders for both criteria, and show that the resulting classical $(L,L-1)$-RACs attain the corresponding proved upper bounds. We also show that these optimal classical codes induce $(L,L-1)$-QRACs that attain a conjectured upper bound on the decoding success probability. Numerical optimization suggests little difference between RACs and QRACs in the average-case setting, but a potentially large classical-quantum gap in the worst-case nonasymptotic regime.

Problem

• Random Access Codes
• Quantum Random Access Codes
• Optimal Constructions
• Decoding Success Probability
• Classical-Quantum Gap

Innovation

• Random Access Codes
• Explicit Construction
• Classical-Quantum Gap
• Optimality
• Worst-case Performance