Skew polycyclic over finite chain rings associated to trinomials
This is an incremental theoretical contribution for coding theorists working on skew polycyclic codes over finite chain rings.
This work studies skew polycyclic codes over finite chain rings defined by central trinomials, introducing an equivalence relation on the defining trinomials that admits a group-theoretic characterization. The authors show that these codes are Hamming equivalent to those defined by a specific trinomial, reducing the classification problem to a canonical case, and determine the size of the equivalence class.
This work studies skew polycyclic codes over finite chain rings defined by central trinomials. For this class of codes, we investigate Hamming equivalence in the non-commutative (skew) setting. We introduce an equivalence relation on the defining trinomials and demonstrate that it admits a group-theoretic characterization in terms of a group of binomials equipped with the Schur multiplication. We determine the conditions under which skew polycyclic codes are Hamming equivalent to those defined by the specific trinomial $x^n-(x^\ell+1)$. This reduces the classification problem for these codes, up to Hamming equivalence, to a canonical case. Finally, we determine the size of the corresponding equivalence class using the decomposition of the unit group of the underlying chain ring.