Quaternary codes and a class of 2-designs invariant under the group

In this paper, we use the Key-Moori Method 1 and construct a quaternary code from a primitive representation of the group of degree 15. We see that is a self-orthogonal even code with the automorphism group isomorphic to the alternating group . It is shown that by taking the support of any codeword of weight in or , and orbiting it under , a 2- design invariant under the group is obtained, where . A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers is determined and moreover, primitivity of on each design is examined.