Algorithms for the Computation of Canonical Forms and Groups of Automorphisms of Linear Codes over Finite Rings and Related Objects

由于发现了比有限域上的码更好的有限环上的码(可以使用相同的比特数来纠正更多的错误)，因此人们对环上的码越来越感兴趣。在提议的项目中，我们希望开发几个算法来处理这样的代码。一个重要的步骤是计算有限链环上线性码的标准型。作为副产品，这也将允许我们计算给定码的自同构群，并且它将允许我们检查两个给定码是否等价。我们还想用这些算法来研究类似的对象，如有限射影几何中的密码函数和点集。有了一个好的算法，我们也将能够对某些代码进行分类，这允许提供一个完整的列表，所有‘不同的’代码。另一个应用是以规范形式给出的环上码的预期数据库，作为我们小组对国际纠错码表的进一步贡献。它将我们的分类方法从域上的码扩展到环上的码，并且应该包含关于这类码的现有知识。此外，我们还希望提供一个关于两两等价的检验，以识别不同结构给出同构对象的情况。

Since the discovery of codes over finite rings which are better than codes over finite fields (it is possible to correct more errors using the same number of bits) there is an increased interest in codes over rings. In the proposed project we want to develop several algorithms to handle such codes. An important step is the computation of a canonical form of a linear code over a finite chain ring. As a byproduct this will also allow us to compute the group of automorphisms of a given code, and it will allow us to check whether two given codes are equivalent. We also want to use these algorithms to study similar objects like cryptographic functions and point-sets in a finite projective geometry. Having at hand a good algorithm we will also be able to classify certain codes, which allows to provide a complete list of all ’different’ codes. One further application is the intended database of codes over rings, given in canonical form, as a further contribution to the international tables of error-correcting codes from our group. It extends our classification methods from codes over fields to codes over rings and should comprise the present knowledge on such codes. We want in addition to provide a test on pairwise equivalence to recognize cases where different constructions give isomorphic objects.