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Keywords:
invariant code; centralizer; affine plane
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
[2] Hughes D. R., Piper F. C.: Projective Planes. Berlin-Heidelberg- New York: Springer Verlag (1973). MR 0333959 | Zbl 0267.50018
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