Representation Theoretic Codes

Abstract

We study the binary and ternary orthogonal codes generated by the weight matrices of 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 the simple Lie algebras of types A and D 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.