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It has been proven in a series of works that the order of the automorphismgroup of a binary [72,36,16] code does not exceed five. We obtain aparametrization of all self-dual binary codes of length 72 with automorphism oforder 4 which can be extremal. We use extensive computations in MAGMA and on asupercomputer to show that an extremal binary code of length 72 does not havean element of order 4.

The Automorphism Group of a Self Dual Binary [72,36,16] Code Does Not Contain Z4