Exploring invariant linear codes through generators and centralizers

Partha Pratim Dey

Department of Computer Science and Engineering, North South University, Dhaka, Bangladesh

E-mail. ppd@northsouth.edu

We investigate a $H$-invariant linear code $C$ over the finite field $F_{p}$ where $H$ is a group of linear transformations. We show that if $H$ is a noncyclic abelian group and $(\vert{H}\vert,p)=1$, then the code $C$ is the sum of the centralizer codes $C_{c}(h)$ where $h$ is a nonidentity element of $H$. Moreover if $A$ is subgroup of $H$ such that $A\cong Z_{q} \times Z_{q}$, $q\neq p$, then $\dim C$ is known when the dimension of $C_{c}(K)$ is known for each subgroup $K\neq 1$ of $A$. In the last few sections we restrict our scope of investigation to a special class of invariant codes, namely affine codes and their centralizers. New results concerning the dimensions of these codes and their centralizers are obtained.

AMSclassification. 05E20.

Keywords.  Invariant code, centralizer, affine plan.