In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain conditions these orbit matrices yield self-orthogonal codes over finite fields $ \mathbb{F}_q $, where $ q $ is a prime power and over finite rings $ \mathbb{Z}_m $. As a case study, we construct codes from orbit matrices of Seidel, Laplacian and signless Laplacian matrices of strongly regular graphs. In particular, we construct self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of the Higman-Sims and McLaughlin graphs.

中文翻译：

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强规则图的Seidel和Laplacian矩阵的轨道矩阵的自正交代码

在本文中，我们介绍了一些强正则图的整数矩阵（例如Seidel和Laplacian矩阵）的轨道矩阵关于其置换自同构群的概念。我们进一步证明，在某些条件下，这些轨道矩阵在有限域$ \ mathbb {F} _q $上产生自正交码，其中$ q $是质数，在有限环$ \ mathbb {Z} _m $上。作为案例研究，我们根据强正则图的Seidel，Laplacian和无符号Laplacian矩阵的轨道矩阵构造代码。特别是，我们从Higman-Sims图和McLaughlin图的Seidel矩阵和Laplacian矩阵的轨道矩阵构造自正交代码。