Year: 2026
Type: preprint
Abstract: Maximum rank-distance codes were introduced independently by Delsarte (1978) and Gabidulin (1985) for unrestricted matrix spaces, and their study was subsequently extended to the classical form spaces of alternating, symmetric, and Hermitian matrices. For Hermitian $d$-codes, Schmidt (2018) established the upper bound and constructed optimal additive codes for all admissible parameters except when $n$ and $d$ are both even and $4\le d\le n-2$. We settle this final open existence case, more generally, construct additive maximum Hermitian rank-metric codes for every admissible length and minimum distance. Rather than working directly in the space of Hermitian matrices, we construct our objects within central simple algebras with unitary involution arising from quotients of skew polynomial rings. We also introduce scalar-field and Lyapunov operators in these algebras, leading to new invariants for the equivalence of additive Hermitian codes. Explicitly determining these invariants yields strong necessary conditions for Hermitian equivalence and demonstrates that our construction contains a large number of pairwise inequivalent codes.
Source: arXiv (Cornell University)
Authors: Santonastaso, Paolo, Zhou, Yue
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3293424821
OpenAlex ID: W7223022355