Binary Codes From Some \(2\)-\((64, 28, 12)\) Designs and their Orbit Matrices

It is known that there are at least 8784 non-isomorphic designs with parameters \(2-(64, 28, 12)\) whose derived \(2-(28, 12, 11)\) designs are quasi-symmetric. In this paper, we examine the binary codes related to a class of non-isomorphic designs with these parameters and invariant under the Frobenius group of order 21 for which the derived \(2-(28, 12, 11)\) designs are not quasi-symmetric. We show that up to equivalence, there are 30 non-isomorphic binary codes obtained from them. Moreover, we classify the self-orthogonal doubly-even codes of length 13 obtained from the non-fixed parts of orbit matrices of these \(2-(64, 28, 12)\) designs under an action of an automorphism group of order four having 12 fixed points. The subcodes of codimension 1 and minimum weight 8 in these codes are all optimal single weight codes.