Outcomes

The project will deal with construction of codes and combinatorial structures, such as combinatorial designs, designs over finite fields (q-ary designs), difference sets and various types of graphs, for example strongly regular graphs and divisible design graphs. Beside linear codes over finite fields and finite rings, we will also investigate quantum codes and network codes. In fact, the links and connections between codes and combinatorial structures are the main focus and emphasis of the proposed project. For constructions of the combinatorial structures and codes we will mainly use algebraic, geometric and enumerative techniques, as well as computational methods. When dealing with combinatorial structures, these include in particular constructions from finite groups and constructions using orbit matrices. The very general Kramer-Mesner method will be implemented and combined with other methods like that of tactical decompositions. Codes will be constructed directly from the obtained combinatorial structures, and vice versa. Our research in scope of this project will also include constructions of Hadamard matrices and related designs and codes. All gotten structures will be analyzed in terms of their automorphism groups and other structural (geometric, algebraic) properties. Obtained codes, designs and graphs will be compared with the previously known structures. As a result of the proposed research, we expect constructions of codes with good properties, possibly better than the presently known codes, and constructions and classifications of various combinatorial structures.