Collaboration in Algebraic Coding Theory - Department Funds

Algebraic coding theory lies at the interface of algebra and information theory. Its purpose is the protection of information against disturbances during transmission and thus has the overarching goal to design powerful codes for various scenarios of information transmission. Depending on the scenario such codes can be classical linear codes over finite fields or codes over certain finite rings, but also subspace codes - each one endowed with a suitable metric. Subspace codes have proven to be the appropriate tool when transmitting information over a network where the nodes transmit random linear combinations of the incoming information vectors in order to maximize the information throughput. Coding theory aims at analyzing and designing codes of the types described above (and many others). My research has two focal points: the study of subspace codes and duality theory for codes over finite rings. Concerning subspace codes, I plan to continue the theory of cyclic subspace codes with a focus on construction and decodability. Another project deals with a particular linkage construction, which I described and studied in an earlier work. The construction shows a lot of potential for the design of codes with strong properties, for instance, with prallelizable decoding schemes. The project aims at investigating the linkage construction with respect to decoding complexity. The third project pertains to rank-metric codes. These are spaces of matrices satisfying a rank condition and possibly some additional structural properties. Via a lifting construction they give rise to subspace codes and, as expected, large rank leads to large distance. The area of rank-metric codes with given Ferrers structure yields intriguing questions at the interplay of combinatorics and linear algebra. In the area of codes over finite rings I plan to continue earlier work by focusing on codes over module alphabets. Duality theory deals with the question as to which type of information about a code fully determines the same type of information for the dual code. MacWilliams duality for module alphabets is closely related to earlier work for codes over groups and it appears promising to extend those results to module alphabets. On the other hand, the MacWilliams extension property, which deals with the question whether weight-preserving isomorphisms between two codes can be extended to a weight-preserving isomorphism on the entire space, surprisingly fails for most cases of codes over module alphabets. Further investigation is needed as to whether additional conditions will guarantee the extension property. This topic is motivated by its close connection to quantum codes, where the error group can be related to a code whose alphabet is a vector space over the binary field. Results in the above-described direction would contribute to the still young theory of stabilizer quantum codes.