The covering radius of Butson Hadamard codes for the homogeneous metric

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.