On the binary self-dual doubly-even codes of length 112

The existence of an extremal binary self-dual doubly-even code of length 112 is an open problem. In this work we study the automorphism group Aut(C) of such a putative code. The types of all the possible automorphisms of odd order of Aut(C) are found: at most 11 of prime and at most 17 of composite order, where the largest prime order is 13. We present a construction of a binary self-dual doubly-even code of length 112 having an automorphism of order 35. An extremal code with such an automorphism does not exist whereas the given construction generates a large number of self-dual [112, 56, 16] codes. As an illustration we list 25 of them.