Let G be the primitive permutation group U₃(4). We enumerate and classify all linear binary codes invariant under U₃(4) from primitive permutation representations of degree 208. We study the properties of codes of small dimensions where computations are possible. We find that up to isomorphism, there are precisely 52 non-trivial binary codes, some projective doubly even self orthogonal codes invariant under this group.

binary codes, design, automorphism group.