The automorphism group of a self-dual [72, 36, 16] code does not contain S3, A4 or D8
نویسندگان
چکیده
A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 whose automorphism group contains the symmetric group of degree 3, the alternating group of degree 4 or the dihedral group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5 or is isomorphic to the elementary abelian group of order 8.
منابع مشابه
On the Automorphism Group of a Binary Self-Dual Doubly Even [72, 36, 16] Code
We prove that the automorphism group of a binary self-dual doubly-even [72, 36, 16] code has order 5, 7, 10, 14 or d where d divides 18 or 24, or it is A4 × C3.
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Abstract. Let C be an extremal self-dual binary code of length 72 and g ∈ Aut(C) be an automorphism of order 2. We show that C is a free F2〈g〉 module and use this to exclude certain subgroups of order 8 of Aut(C). We also show that Aut(C) does not contain an element of order 10. Combining these results with the ones obtained in earlier papers we find that the order of Aut(C) is either 5 or divi...
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The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6 . We show that C , as an F2〈g〉 -module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find...
متن کاملThe Automorphism Group of an Extremal {72, 36, 16} Code Does Not Contain Z7, Z3×Z3, or D10
A computer calculation with Magma shows that there is no extremal self-dual binary code C of length 72 that has an automorphism group containing either the dihedral group of order 10, the elementary abelian group of order 9, or the cyclic group of order 7. Combining this with the known results in the literature one obtains that the order of Aut(C) is either 5 or divides 24.
متن کاملThe Automorphism Group of a Binary Self-Dual Doubly Even [72, 36, 16] Code is Solvable
We prove that the automorphism group of a putative binary self-dual doublyeven [72,36,16] code is solvable. Moreover, its order is 5, 7, 10, 14, 56, or a divisor of 72.
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عنوان ژورنال:
- Adv. in Math. of Comm.
دوره 7 شماره
صفحات -
تاریخ انتشار 2013