The trace dual of nonlinear skew cyclic codes
Daniel Bossaller, Daniel Herden, Indalecio Ruiz-Bolaños
TL;DR
The paper addresses the trace duals of $F_q$-linear over $F_{q^2}$ codes, focusing on both cyclic and skew cyclic structures under trace Euclidean and trace Hermitian inner products. It develops explicit generator descriptions by factoring $X^n-1$ as $w\ell f g$ and, where needed, employing gcd-based constructions, while also treating skew cyclic codes via the skew polynomial ring and projection to $F_q$-codes. The main contributions include explicit formulas for $\mathcal{C}^{\perp_{TE}}$ and $\mathcal{C}^{\perp_{TH}}$ in the cyclic and skew cyclic settings, generalizing Verma-Sharma's results and removing restrictive gcd assumptions. This work provides constructive duals that enhance the design of nonlinear and quantum-error-correcting codes and lays groundwork for extending the framework to broader field extensions and distance analyses.
Abstract
Codes which have a finite field $\mathbb{F}_{q^m}$ as their alphabet but which are only linear over a subfield $\mathbb{F}_q$ are a topic of much recent interest due to their utility in constructing quantum error correcting codes. In this article, we find generators for trace dual spaces of different families of $\mathbb{F}_q$-linear codes over $\mathbb{F}_{q^2}$. In particular, given the field extension $\mathbb{F}_q\leq \mathbb{F}_{q^2}$ with $q$ an odd prime power, we determine the trace Euclidean and trace Hermitian dual codes for the general $\mathbb{F}_q$-linear cyclic $\mathbb{F}_{q^2}$-code. In addition, we also determine the trace Euclidean and trace Hermitian duals for general $\mathbb{F}_q$-linear skew cyclic $\mathbb{F}_{q^2}$-codes, which are defined to be left $\mathbb{F}_q[X]$-submodules of $\mathbb{F}_{q^2}[X;σ]/(X^n-1)$, where $σ$ denotes the Frobenius automorphism and $\mathbb{F}_{q^2}[X;σ]$ the induced skew polynomial ring.