# Article

**Keywords:**

invariant code; centralizer; affine plane

**Summary:**

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 $A\cong Z_{q} \times Z_{q}$, $q\ne p$, then dim $C$ is known when the dimension of $C_{c}(K)$ is known for each subgroup $K\ne 1$ 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.

References:

[1] Hall M.:

**Combinatorial Theory**. New York-Chichester-Brisbane-Toronto- Singapore: Interscience (1986).

MR 0840216 |

Zbl 0588.05001
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**Projective Planes**. Berlin-Heidelberg- New York: Springer Verlag (1973).

MR 0333959 |

Zbl 0267.50018