Exploring invariant linear codes through generators and centralizers
Archivum mathematicum, Tome 41 (2005) no. 1, pp. 17-26
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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.
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.
Dey, Partha Pratim. Exploring invariant linear codes through generators and centralizers. Archivum mathematicum, Tome 41 (2005) no. 1, pp. 17-26. http://geodesic.mathdoc.fr/item/ARM_2005_41_1_a2/
@article{ARM_2005_41_1_a2,
author = {Dey, Partha Pratim},
title = {Exploring invariant linear codes through generators and centralizers},
journal = {Archivum mathematicum},
pages = {17--26},
year = {2005},
volume = {41},
number = {1},
mrnumber = {2142140},
zbl = {1115.05097},
language = {en},
url = {http://geodesic.mathdoc.fr/item/ARM_2005_41_1_a2/}
}