Type IV codes over a non-unital ring

Abstract

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.