The polycyclic codes over the finite field $ \mathbb{F}_q $

Abstract

This article extended the properties of the idempotent generator of cyclic codes to polycyclic codes over the finite field $ \mathbb{F}_q $. In addition, the check matrix of polycyclic codes was provided over $ \mathbb{F}_q $. Specifically, it has been proven that the constacyclic code is an $ {{{\rm{MDS}}}} $ code over $ \mathbb{F}_q $ if and only if its annihilator dual code is also an $ {{{\rm{MDS}}}} $ code. Finally, we have provided some examples of good codes.

Keywords

• polycyclic codes
• idempotent generator
• check matrix
• $ {{{\rm{mds}}}} $ codes