The aim of this work is to study the structure and properties of the binary LCD codes having an automorphism of odd prime order and to present a method for their construction.

We investigate right ideals as codes in twisted group algebras. Such codes are called twisted group codes. It turns out that many interesting codes belong to this class; for instance, the ternary extended Golay code, Hamming codes and constacyclic codes. In particular we characterize all linear codes which are twisted group codes in terms of their automorphism group.

Let$C$be a binary extremal self-dual code of length 96. We prove that an automorphism of$C$of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then${\rm Aut}(C)$is solvable and its order divides$2^{5}3$or$2^{5}5$or${\rm Aut}(C)$is the alternating group${\rm A}_{5}$which is the only possible group of order 60. Furthermore,$\vert {\rm Aut}(C)\vert = 20$or$40$cannot occur.