A new class of codes over Z₂ x Z₂

We study a new class of codes over Z_2 x Z_2 which we call L-codes. They arise as a natural fifth step in a series of analogies between Kleinian codes, binary codes, lattices and vertex operator algebras. This analogy will be explained in detail. We classify self-dual L-codes up to length 10 and provide tables for these codes and their weight enumerators up to length 4. We also discuss extremal codes for which a nearly complete classification is obtained.