A polymatroid approach to generalized weights of rank metric codes

Sudhir R Ghorpade,Trygve Johnsen

doi:10.1007/s10623-020-00798-9

Abstract

We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.

Highlights

Rank metric codes are an important variant of linear codes, and they have gained prominence in the past few decades, partly due to myriad applications in network coding and cryptography, as due to their intrinsic interest
A little more general class of (q, m)-demi-polymatroids, define generalized weights for them, and establish a duality in this context. We show that these can be applied to flags, or chains, of Delsarte rank metric codes
These results are inspired by the related results of Ravagnani [18] and of Martínez-Peñas and Matsumoto [15] about the generalized weights and the relative generalized weights of Delsarte rank metric codes

Summary

Introduction

Rank metric codes are an important variant of linear (block) codes, and they have gained prominence in the past few decades, partly due to myriad applications in network coding and cryptography, as due to their intrinsic interest. A little more general class of (q, m)-demi-polymatroids, define generalized weights for them, and establish a duality in this context We show that these can be applied to flags, or chains, of Delsarte rank metric codes. One could ask for some “m-fold” version of Wei duality, and that is what we give in Theorems 15 and 17 below These results are inspired by the related results of Ravagnani [18] and of Martínez-Peñas and Matsumoto [15] about the generalized weights and the relative generalized weights of Delsarte rank metric codes. The following result may be viewed as a version of Wei duality for generalized weights of (q, m)-demi-polymatroids. The proof of Theorem 17 uses arguments that are analogous to those in the proof of [18, Corollary 38]

Generalized weights of flags of Delsarte rank metric codes

Demi-polymatroids associated to flags of Delsarte codes

Generalized weights of flags of Delsarte codes

Duality of Delsarte rank metric codes

Duality for flags of Delsarte rank metric codes

Another definition of generalized weights