Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes

TL;DR

This paper explores the weight hierarchy of Grassmann and Schubert codes, introducing subclose families as a new combinatorial tool linked to threshold graphs to advance understanding of these codes.

Contribution

It generalizes the concept of closed families to subclose families and investigates their properties and connections to threshold graphs, aiding in code analysis.

Findings

Subclose families are a useful generalization of closed families.

Connections between subclose families and threshold graphs are established.

Partial solutions to the weight hierarchy problem for Grassmann codes are discussed.

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. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a closed 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.