Hermitian rank-distance codes constructed using skew polynomial rings
New Maximum Hermitian $d$-Codes and New Invariants from Skew Polynomial Rings
Information Theory
Summary
Hermitian rank-distance codes are special arrangements of matrices used in error correction and data storage. Before this work, some configurations of these codes were not known to exist, especially when certain parameters were even numbers. The authors solved this last piece by building these codes in a new mathematical space called skew polynomial rings, which helps manage complex symmetries. They also introduced new tools to tell when two codes are essentially the same or different, showing their solutions offer many unique options.
What this means in practice
- •For data storage engineers: Construct optimal rank-distance codes for reliable error correction in storage systems needing Hermitian matrix structures.
- •For wireless communication developers: Implement new classes of Hermitian codes for efficient and robust signal encoding and decoding in MIMO systems.
Authors
Paolo Santonastaso, Yue Zhou
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.