New Maximum Hermitian d-Codes and New Invariants from Skew Polynomial Rings
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≤ d≤ 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.
Related
- Hermitian K-theory for stable ∞-categories III: Grothendieck--Witt groups of rings
- On classical and Gorenstein homological invariants of rings
- Polynomial Invariants of Pseudo-Anosov Maps (Addendum)
- Welschinger invariants and the Conway polynomial
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
BibTeX
@misc{santonastaso2026maximum,
title = {{New Maximum Hermitian $d$-Codes and New Invariants from Skew Polynomial Rings}},
author = {Paolo Santonastaso and Yue Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26048},
eprint = {2610.26048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26048},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - New Maximum Hermitian d-Codes and New Invariants from Skew Polynomial Rings
AU - Paolo Santonastaso
AU - Yue Zhou
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26048
UR - https://arxiv.org/abs/2610.26048
AB - 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≤ d≤ 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.
ER -