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

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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.