Characterizations and properties of principal $(f, \sigma, \delta)$-codes over rings

Let $A$ be a ring with identity, $\sigma$ a ring endomorphism of $A$ that maps the identity to itself, $\delta$ a $\sigma$-derivation of $A$, and consider the skew-polynomial ring $A[X;\sigma,\delta]$. When $A$ is a finite field, a Galois ring, or a general ring, some fairly recent literature used $A[X;\sigma,\delta]$ to construct new interesting codes (e.g. skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g. cyclic and constacyclic linear codes). This paper presents results concerning {\it principal} $(f, \sigma, \delta)$-codes over a ring $A$, where $f\in A[X;\sigma,\delta]$ is monic. We provide recursive formulas that compute the entries of both a generating matrix and a control matrix of such a code $\mathcal{C}$. When $A$ is a finite commutative ring with identity and $\sigma$ is a ring automorphism of $A$, we also give recursive formulas for the entries of a parity-check matrix of $\mathcal{C}$. Also in this case, with $\delta=0$, we give a generating matrix of the dual $\mathcal{C}^\perp$, present a characterization of principal $\sigma$-codes whose duals are also principal $\sigma$-codes, and deduce a characterization of self-dual principal $\sigma$-codes. Some corollaries concerning principal $\sigma$-constacyclic codes are also given, and some highlighting examples are provided.