Perfect Codes in the Johnson Schemes

Abstract

In his pioneering work, from 1973, on algebraic approach to codes in association schemes, Dlesarte has conjectured that there are no nontrivial perfect codes in the Johnson schemes. Many attempts were made during the last 30 years to solve this conjecture. These attempts used Lloyd polynomials, anticodes in the Johnson schemes, designs, and number theory. We will survey all the known results and outline directions for solving the problem.