The covering radius of Butson Hadamard codes for the homogeneous metric

TL;DR

This paper investigates the covering radius of Butson Hadamard codes within the homogeneous metric, deriving bounds through orthogonal array arguments and bent sequences, extending previous Hamming metric results.

Contribution

It introduces bounds for the covering radius of Butson Hadamard codes in the homogeneous metric, generalizing existing Hamming metric bounds using new algebraic and combinatorial methods.

Findings

An upper bound on the covering radius is established.

A lower bound based on bent sequences is derived.

The bounds generalize previous results for the Hamming metric.

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.