A Constructive Method to Minimize the Index of Coincidence under Marginal Constraints

We consider the problem of minimizing the index of coincidence of a joint distribution under fixed marginal constraints. This objective is motivated by several applications in information theory, where the index of coincidence naturally arises. A closed-form solution is known when the marginals satisfy a strong feasibility condition, but this condition is rarely met in practice. We first show that the measure of the set of marginals for which condition applies vanishes as the dimension grows. We then characterize the structure of the optimal coupling in the general case, proving that it exhibits a monotone staircase of zero entries. Based on this structure, we propose an explicit iterative construction and prove that it converges in finitely many steps to a minimizer. Main result of the paper is a complete constructive solution of index-of-coincidence minimization.