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Exploring invariant linear codes through generators and
centralizers

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*Partha Pratim Dey*

**Address.**

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

**E-mail. **ppd@northsouth.edu

**Abstract.**

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.