Subclose families, threshold graphs, and the weight hierarchy of Grassmann and Schubert Codes

Abstract

We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and. more generally, to Schubert varieties in Grassmannians. In geometric terms, this corresponds to the determination of the maximum number of F.-rational points on sections of Schubert varieties (with nondegenerate Plucker embedding) by linear subvarieties of a fixed (co)dimension. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a close family. We propose a generalization of this to what is called a subclose family. A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined.