Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding

TL;DR

This paper introduces new parameters called RDIP and RGRW for linear codes, analyzes their properties, and applies them to design secure network coding schemes with guaranteed security and error correction capabilities.

Contribution

It defines the relative generalized rank weight and related parameters, clarifies their properties, and provides an explicit construction of a secure network coding scheme solving an open problem.

Findings

Established the relation between RGRW and minimum rank distance.

Analyzed security and error correction of the proposed coding scheme.

Provided an explicit construction of a secure network coding scheme.

Abstract

By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C_1 of length n over a field extension and its subcode C_2. One is called the relative dimension/intersection profile (RDIP), and the other is called the relative generalized rank weight (RGRW). We clarify their basic properties and the relation between the RGRW and the minimum rank distance. As applications of the RDIP and the RGRW, the security performance and the error correction capability of secure network coding, guaranteed independently of the underlying network code, are analyzed and clarified. We propose a construction of secure network coding scheme, and analyze its security performance and error correction capability as an example of applications of the RDIP and the RGRW. Silva and Kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim C_1-1 links are wiretapped, which is guaranteed over any underlying network code. However, the explicit construction of such a scheme remained an open problem. Our new construction is just one instance of secure network coding that solves this open problem.