Adjoint and duality for rank-metric codes in a skew polynomial framework
José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro, Paolo Santonastaso
TL;DR
This work develops an explicit adjoint and duality theory for rank-metric codes arising from quotients of skew polynomial rings $R=\mathbb{F}_{q^n}[x;\sigma]$ by $RF(x^n)$. By identifying $R_F=R/RF(x^n)$ with the matrix algebra $M_n(\mathbb{F}_{q^s})$ (where $s=\deg(F)$) and using the reciprocal polynomial $\hat{F}$, the authors construct a skew-polynomial anti-isomorphism $\Theta:R_F\to R_{\hat{F}}$ that encodes transposition via $M_{R_F}(a)^T= M_{R_{\hat{F}}}(\Theta(a))$ up to conjugation. They then establish a Frobenius-algebra-based duality, linking the rank-metric dual to a corresponding adjoint under a Frobenius form, and derive explicit duals for key MRD families: the $S_{n,s,k}(\eta,\rho,F)$ and $D_{n,s,k}(\gamma,F)$ codes. These results allow computation of idealisers, centralisers, and centres, and yield new MRD codes for infinitely many parameter sets, extending previous constructions and providing a systematic algebraic framework for classifying and extending MRD codes in the skew-polynomial setting.
Abstract
Skew polynomial rings provide a fundamental example of noncommutative principal ideal domains. Special quotients of these rings yield matrix algebras that play a central role in the theory of rank-metric codes. Recent breakthroughs have shown that specific subsets of these quotients produce the largest known families of maximum rank distance (MRD) codes. In this work, we present a systematic study of transposition and duality operations within quotients of skew polynomial rings. We develop explicit skew-polynomial descriptions of the transpose and dual code constructions, enabling us to determine the adjoint and dual codes associated with the MRD code families recently introduced by Sheekey et al. Building on these results, we compute the nuclear parameters of these codes, and prove that, for a new infinite set of parameters, many of these MRD codes are inequivalent to previously known constructions in the literature.