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Erschienen in:
01.08.2013 | Original Paper
On the automorphism group of a binary self-dual \([120, 60, 24]\) code
Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 3-4/2013
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Abstract
We prove that an automorphism of order 3 of a putative binary self-dual \([120, 60, 24]\) code \(C\) has no fixed points. Moreover, the order of the automorphism group of \(C\) divides \(2^a\cdot 3 \cdot 5\cdot 7\cdot 19\cdot 23\cdot 29\) with \(a\in \mathbb N _0\). Automorphisms of odd composite order \(r\) may occur only for \(r=15, 57\) or \(r=115\) with corresponding cycle structures \(3 \cdot 5\)-\((0,0,8;0), 3\cdot 19\)-\((2,0,2;0)\) or \(5 \cdot 23\)-\((1,0,1;0)\) respectively. In case that all involutions act fixed point freely we have \(|\mathrm{Aut}(C)| \le 920\), and \(\mathrm{Aut}(C)\) is solvable if it contains an element of prime order \(p \ge 7\). Moreover, the alternating group \(\mathrm{A}_5\) is the only non-abelian composition factor which may occur in \(\mathrm{Aut}(C)\).