An extremal [72, 36, 16] binary code has no automorphism group containing Z2 x Z4, Q8, or Z10
نویسنده
چکیده
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 divides 24. If 8 divides the order of Aut(C) then its Sylow 2-subgroup is either D8 or Z2 × Z2 × Z2.
منابع مشابه
The 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 Self-Dual [72, 36, 16] Binary Code Does Not Contain Elements of Order 6
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 a self-dual [72, 36, 16] code does not contain S3, A4 or D8
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.
متن کاملThe automorphism group of a self-dual binary [72,36,16] code does not contain Z7, Z3xZ3, 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 D10 of order 10, the elementary abelian group Z3 ×Z3 of order 9, or the cyclic group Z7 of order 7. Combining this with the known results in the literature one obtains that Aut(C) is either Z5 or has order dividing 24.
متن کاملOn the Automorphism Group of a Binary Self-Dual Doubly Even [72, 36, 16] Code
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ورودعنوان ژورنال:
- Finite Fields and Their Applications
دوره 18 شماره
صفحات -
تاریخ انتشار 2012