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

有限环及相关对象上线性码的正则形式和自同构群的计算算法

• 批准号: 171110320
• 负责人: Professor Dr. Alfred Wassermann
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171110320?language=en
• 关键词: Algorithms; Computation; Canonical; Forms; Groups

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.

由于发现了有限环上的代码比有限域上的代码更好（使用相同数量的位可以纠正更多的错误），因此对环上的代码的兴趣增加。在拟议的项目中，我们要开发几种算法来处理这样的代码。一个重要的步骤是计算一个标准型的线性码在有限链环。作为副产品，这也将允许我们计算给定代码的自同构群，并允许我们检查两个给定代码是否等价。我们也希望使用这些算法来研究类似的对象，如加密函数和有限射影几何中的点集。手头有一个好的算法，我们也将能够对某些代码进行分类，这允许提供所有“不同”代码的完整列表。另一个应用是环上代码的数据库，以规范形式给出，作为我们小组对国际纠错码表的进一步贡献。它扩展了我们的分类方法，从代码领域的代码环，并应包括目前的知识，这样的代码。此外，我们还想提供一个成对等价的测试，以识别不同构造给出同构对象的情况。