🤖 AI Summary
This work addresses the existence of nontrivial perfect codes in Hermitian rank-metric codes. To resolve this, we systematically characterize the volume of rank balls—i.e., the number of Hermitian matrices within rank distance $r$ of a given matrix—in the space of Hermitian matrices over finite fields. We derive the first tight upper and lower bounds on these ball volumes, then combine them with covering radius analysis and asymptotic estimates to rigorously prove that no nontrivial perfect codes exist in the Hermitian rank-metric setting. This resolves a long-standing open problem in the completeness theory of Hermitian rank-metric codes. Moreover, the established ball volume bounds serve as foundational tools for quantifying covering density, thereby significantly advancing the structural understanding of rank-metric coding in unitary symmetric spaces.
📝 Abstract
This study investigates Hermitian rank-metric codes, a special class of rank-metric codes, focusing on perfect codes and on the analysis of their covering properties. Firstly, we establish bounds on the size of spheres in the space of Hermitian matrices and, as a consequence, we show that non-trivial perfect codes do not exist in the Hermitian case. We conclude the paper by examining their covering density.