Polycyclic codes as invariant subspaces
Polycyclic codes of length \(n\) over a finite field \(\mathbb{F}\) can be described as ideals in the quotient ring of \(\mathbb{F}[x]/(f),\) where \(f\) is some polynomial of degree \(n.\) The main idea in this work is to consider the codes in the class as invariant subspaces under the action of an endomorphism which can allow some advanced techniques from linear algebra to help solving the problem. The authors apply a polyshift transform that admits for a matrix the companion matrix of the polynomial \(f(x)\). A notion of minimal invariant subspace is introduced, driven by the factorization of \(f(x)\) into irreducible factors over the base field. This allows to describe any invariant subspace as a direct sum of its intersections with these spaces. Similarly, an analogue of the idempotent of cyclic codes is developed and a new bound on the minimum distance is derived which improves, in some cases, the previously established BCH-like bound. The paper concludes with some open problems.
scientific article; zbMATH DE number 7312736
scientific article