Grassmannian Codes with New Distance Measures for Network Coding

TL;DR

This paper introduces new distance measures for Grassmannian codes used in network coding, demonstrating their equivalence, generalization of existing codes, and potential for more efficient network coding solutions.

Contribution

It proposes two generalized distance measures for Grassmannian codes, proves their equivalence, and shows how they extend existing codes and designs for improved network coding performance.

Findings

New distance measures generalize subspace distance

Codes with new measures outperform existing codes in some networks

Hamming scheme codes are a subfamily of Grassmannian codes

Abstract

Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this work we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measureshave minimal requirements for network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts it is proved that codes in the Hamming scheme form a subfamily of the Grassmannian codes.