Representations of Lie Algebras and Coding Theory

TL;DR

This paper explores how weight matrices from simple Lie algebras generate orthogonal error-correcting codes with large minimal distances, linking Lie theory with coding theory.

Contribution

It introduces new constructions of orthogonal codes from Lie algebra modules and analyzes their properties using Weyl groups and representation theory.

Findings

Certain Lie algebra weight matrices generate codes with large minimal distances.

Doubly-even binary and ternary codes are constructed from $sl(n)$ and $o(2n)$.

Codes from $F_4$, $E_6$, $E_7$, $E_8$ modules also have large minimal distances.

Abstract

Linear codes with large minimal distances are important error correcting codes in information theory.Orthogonal codes have more applications in the other fields of mathematics. In this paper, we study the binary and ternary orthogonal codes generated by the weight matrices on finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically. It turns out that certain weight matrices of $sl(n,\mbb{C})$ and $o(2n,\mbb{C})$ generate doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prove that the weight matrices of $F_4$, $E_6$, $E_7$ and $E_8$ on their minimal irreducible modules and adjoint modules all generate ternary orthogonal codes with large minimal distances. In determining the minimal distances, we have used the Weyl groups and branch rules of the irreducible representations of the related simple Lie algebras.