Skew Laurent Series and General Cyclic Convolutional Codes
José Gómez-Torrecillas, José Patricio Sánchez-Hernández
TL;DR
The paper develops a general algebraic framework to extend cyclic convolutional codes from polynomial coefficients to a skew-derivation setting $(\sigma,\delta)$ on a finite-dimensional $\mathbb{F}$-algebra $A$. It introduces the ring of left skew Laurent series $A(\!(X;\sigma,\delta)\!)$ under suitable existence conditions and builds a corresponding module theory, enabling a bijection between $(\sigma,\delta,A)$-cyclic convolutional codes in $\mathbb{F}^n(\!(X)\!)$ and $\mathbb{F}[X]$-direct summands of $\mathbb{F}^n[X]$ that are right $A[X;\sigma,\delta]$-submodules. Under nilpotence assumptions on $\delta$ and $\delta'=-\delta\sigma^{-1}$ (with $\sigma$ an automorphism), these codes correspond to Lopez–Szabó right ideal codes, and, in the semisimple case, admist idempotent generators yielding minimal encoders and distance insights. The construction avoids Noetherian or topological constraints, connects polynomial and Laurent frameworks, and has potential implications for cryptographic constructions based on quasi-cyclic codes.
Abstract
Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible.