Automorphism groups of cyclic codes (Q2269529)

The automorphism group \(\text{Aut}\, {\mathcal C}\) of a binary code \({\mathcal C}\subseteq \mathbb F_2^{\,n}\) is defined to be the subgroup of the symmetric group \(S_n\) operating on the coordinate set and mapping \(\mathcal C\) into itself. The authors give a partial answer to the question which finite groups arise as the automorphism group of a binary cyclic code. They show that \(\text{Aut}\, {\mathcal C}\) cannot be isomorphic to a non-trivial cyclic group of odd order or to an alternating group \({\mathcal A}_n\) of degree \(n\in\{ 3,4,5,6,7\}\) or \(n\geq 9\). (The group \({\mathcal A}_8\) occurs as automorphism group of a binary cyclic code of length 15). For the iterated wreath product \(G=S_{n_1}\wr\ldots \wr S_{n_r}\) for \(r\in\mathbb N\) and \(n_1,\ldots,n_r\in\mathbb N\), \(n_i\geq 3\), there exists a binary cyclic code of length \(n_1\cdots n_r\) admitting \(G\) as automorphism group. This and the following result are shown by explicit construction. For \(a,b\in \mathbb N\) with \(2<a<b\) and \(\text{gcd}(a,b)=1\), there exist binary cyclic codes \({\mathcal C}={\mathcal C}_0(a,b)\) with \(\text{Aut}\, {\mathcal C}\cong S_a \times S_b\). Some of these codes were studied by \textit{J. D. Key} and \textit{P. Seneviratne} [Eur. J. Comb. 28, No. 1, 121--126 (2007; Zbl 1105.94018)] in the context of regular lattice graphs and permutation decoding. A unified treatment of a related code family \({\mathcal C}_1(a,b)\) of binary linear codes is provided. Moreover using known results on permutation groups, the \textit{primitive} permutation groups \(G\leq S_n\) that occur as the automorphism group of a binary cyclic code are characterized; one of the following holds (where cases (2) to (4) do occur). {\parindent=7mm \begin{itemize}\item[(1)]\({\mathcal C}_p \lneqq G \lneqq \text{AGL}(1,p)\), \(p=n\geq 5\) a prime, \item[(2)]\(G\cong S_n\), \item[(3)]\(G\cong \text{P}\Gamma \text{L}(d,q)\) with \(d\geq 3\), \(q=2^k\) and \(n=(q^d-1)/(q-1)\), or \item[(4)]\(G\cong M_{23}\) and \(n=23\). \end{itemize}} Many selected examples of automorphism groups of binary codes and the parameters \((n,k,d)\) of these are given, as well results concerning subgroups of AGL\((1,p)\) for \(n=p\), \(p\) prime, \(5\leq p\leq 79\) (based on computer calculations by the first author as part of his Ph.D.\ thesis).