A new class of codes over Z_2 x Z_2

TL;DR

This paper introduces L-codes over Z_2 x Z_2, explores their classification, and draws analogies with other mathematical structures, providing tables and classifications for various lengths.

Contribution

It presents the first classification of self-dual L-codes up to length 10 and develops a framework linking these codes to lattices and vertex operator algebras.

Findings

Classified self-dual L-codes up to length 10

Provided tables for codes and weight enumerators up to length 4

Achieved nearly complete classification of extremal codes

Abstract

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.