# Article

 Title: Exploring invariant linear codes through generators and centralizers (English) Author: Dey, Partha Pratim Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 41 Issue: 1 Year: 2005 Pages: 17-26 Summary lang: English . Category: math . 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. (English) Keyword: invariant code Keyword: centralizer Keyword: affine plane MSC: 05E20 MSC: 94B05 idZBL: Zbl 1115.05097 idMR: MR2142140 . Date available: 2008-06-06T22:45:00Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107932 . Reference: [1] Hall M.: Combinatorial Theory.New York-Chichester-Brisbane-Toronto- Singapore: Interscience (1986). Zbl 0588.05001, MR 0840216 Reference: [2] Hughes D. R., Piper F. C.: Projective Planes.Berlin-Heidelberg- New York: Springer Verlag (1973). Zbl 0267.50018, MR 0333959 .

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