Representations of Lie Algebras and Coding Theory

2009/02/17 by Xiaoping Xu, Xu, Xiaoping
Mathematics · #17B10 #94B60 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0902.2837

openalex publication_date 2009/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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,\mbbC) and o(2n,\mbbC) 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.

Citations

• Polynomial Representation of F₄ and a New Combinatorial Identity about Twenty-Four
• The children of the (32, 16) doubly even codes

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