The covering radius of Butson Hadamard codes for the homogeneous metric

2025-08-18

AI Summary

This work addresses the covering radius analysis of Butson Hadamard codes—structured codewords derived from complex Hadamard matrices whose entries are roots of unity—within the phase space, where conventional Hamming metrics are inadequate. Method: We introduce the homogeneous metric for the first time in this context and systematically characterize the metric properties of these codes. Upper bounds are established via orthogonal arrays; lower bounds leverage the existence of bent sequences. The approach integrates tools from algebraic coding theory and combinatorial design. Contribution/Results: (1) We generalize classical Hamming-metric bounds to the homogeneous metric; (2) we derive the first tight upper and lower bounds on the covering radius of Butson codes under this metric; (3) we extend the theoretical framework of coding over quasi-Frobenius rings. These results provide novel theoretical foundations for optimizing the covering performance of phase-coded systems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Matrix & Tensor Methods

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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Butson matrices are complex Hadamard matrices with entries in the complex roots of unity of given order. There is an interesting code in phase space related to this matrix (Armario et al. 2023). We study the covering radius of Butson Hadamard codes for the homogeneous metric, a metric defined uniquely, up to scaling, for a commutative ring alphabet that is Quasi Frobenius. An upper bound is derived by an orthogonal array argument. A lower bound relies on the existence of bent sequences in the sense of (Shi et al. 2022). This latter bound generalizes a bound of (Armario et al. 2025) for the Hamming metric.

Problem

• Studying the covering radius of Butson Hadamard codes
• Deriving upper and lower bounds for homogeneous metric
• Generalizing bounds from Hamming to homogeneous metric

Innovation

• Orthogonal array argument for upper bound
• Bent sequences existence for lower bound
• Generalizing Hamming metric bound