
Journal of Lie Theory 22 (2012), No. 3, 647682 Copyright Heldermann Verlag 2012 Representations of Lie Algebras and Coding Theory Xiaoping Xu Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, P. R. China xiaoping@math.ac.cn We present a connection between binary and ternary orthogonal codes and finitedimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras are symmetries of the related codes. It turns out that certain weight matrices of sl(n,C) and o(2n,C) generate doublyeven 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. Keywords: Simple Lie algebra, irreducible module, weight matrix, orthogonal code, minimal weight of a code. MSC: 17B10, 94B60; 17B25 [ Fulltextpdf (406 KB)] for subscribers only. 