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Journal of Lie Theory 22 (2012), No. 3, 647--682
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

We present a connection between binary and ternary orthogonal codes and finite-dimensional 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 doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prove that the weight matrices of F4, E6, E7 and E8 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

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