Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes
Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes
Sudhir R. Ghorpade
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India
E-Mail: srg@math.iitb.ac.in
Arunkumar R. Patil
Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India
E-Mail: arun.iitb@gmail.com
and
Harish K. Pillai
Department of Electrical Engineering
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India
E-Mail: hp@ee.iitb.ac.in
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.
| 1 | Introduction | 1 |
| 2 | Close Families and Subclose Families | 2 |
| 3 | Higher Weights of Grassman Codes and Schubert Codes | 7 |
| 4 | Threshold Graphs, Optimal Graphs end Subclose Families | 9 |
| References | 11 |
This paper is published in Contemporary Mathematics, Vol. 487,
American Mathematical Society, Providence, RI, 2009, pp. 87-99.
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