Lecture Content：

Completely regular codes form a class of well-structured combinatorial configurations. Since their introduction by P. Delsarte, they have been primarily studied in classical distance-regular graphs such as Hamming and Johnson graphs, which are known to be geometric. We discuss how completely regular codes with minimum eigenvalue in geometric graphs can be reduced to simpler completely regular codes in the clique graphs of these graphs. This reduction is then used to resolve several open problems concerning completely regular codes in the Johnson graphs J(n,3) and J(n,4), as well as in Grassmann and Hamming graphs.

Speaker Introduction:

van Mogilnykh is a Russian mathematician with research interests in coding theory, algebraic combinatorics, and related areas. In 2010, he received a Ph.D. in Discrete Mathematics and Theoretical Cybernetics from the Sobolev Institute of Mathematics. In 2013, he was a visiting researcher at the Universitat Autònoma de Barcelona, Spain. He currently works at the Sobolev Institute of Mathematics in Novosibirsk.

I.M. has authored over 30 scientific papers on the structure and spectral properties of codes, block designs, and graphs, published in journals including Designs, Codes and Cryptography, Discrete Mathematics, and Problems of Information Transmission. Among his main research contributions are: solving the existence problem of perfect bitrades in the Hamming scheme in collaboration with F. Solov'eva; establishing lower bounds on the sizes of clique trades in Grassmann graphs together with D. Krotov and V. Potapov; and characterizing Cameron-Liebler line classes in PG(n,4) in joint work with A. Gavrilyuk.