There is a special local ring E of order 4, without identity for the multiplication, defined by E=⟨a,b∣∣2a=2b=0,a2=a,b2=b,ab=a,ba=b⟩. We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over E, and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.