Classification of self-dual codes of length 50 with an automorphism of odd prime order

Abstract

By applying a method for constructing binary self-dual codes with an automorphism of odd prime order $$p$$ p , we give a full classification of all optimal binary self-dual codes of length 50 having an automorphism of order 3. As a consequence, we give a full classification of all $$[50, 25, 10]$$ [ 50 , 25 , 10 ] codes possessing an automorphism of odd prime order. Up to equivalence, there are exactly 177,601 such codes. This completely determines all possibilities for the cardinality of the automorphism group of such a code. Also, we show that there are at least 52 non-isomorphic quasi-symmetric 2-(49, 9, 6) designs, derived from the $$[50,25,10]$$ [ 50 , 25 , 10 ] codes with a particular weight enumerator.