Geodesic
The automorphism group of a~$q$-ary Hamming code
Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 6, pp. 50-55

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It is well known that the semilinear symmetry group of a $q$-ary Hamming code $\mathcal H$ with length $n=\frac{q^m-1}{q-1}$ is isomorphic to $\mathit\Gamma L_m(q)$. This does not clarify if all symmetries of the code are semilinear or not. Here we prove that each symmetry of the code constituted by all triples in $\mathcal H$ is semilinear. This implies that every symmetry of the Hamming code is semilinear. So, it is shown that the automorphism group of a $q$-ary Hamming code is isomorphic to the semidirect product $\mathit\Gamma L_m(q)\rightthreetimes\mathcal H$. Bibliogr. 4.
Keywords: the Hamming code
Mots-clés : automorphism group. 
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     author = {E. V. Gorkunov},
     title = {The automorphism group of a~$q$-ary {Hamming} code},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {50--55},
     publisher = {mathdoc},
     volume = {17},
     number = {6},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2010_17_6_a2/}
}
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E. V. Gorkunov. The automorphism group of a~$q$-ary Hamming code. Diskretnyj analiz i issledovanie operacij, Tome 17 (2010) no. 6, pp. 50-55. http://geodesic.mathdoc.fr/item/DA_2010_17_6_a2/