A Rank-Metric Approach to Error Control in Random Network Coding

TL;DR

This paper introduces a rank-metric approach to error control in random network coding, linking subspace codes to rank-metric codes, and proposes efficient decoding algorithms that leverage erasures and deviations for improved error correction.

Contribution

It demonstrates how subspace codes can be constructed from rank-metric codes and reformulates decoding as a generalized problem, enhancing error correction capabilities in network coding.

Findings

Codes can be constructed from rank-metric codes preserving distance.

Decoding can incorporate erasures and deviations for better correction.

Efficient decoding algorithms for Gabidulin codes are proposed.

Abstract

The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is investigated. It is shown that codes in this class can be easily constructed from rank-metric codes, while preserving their distance properties. Moreover, it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank-metric codes where partial information about the error is available. This partial information may be in the form of erasures (knowledge of an error location but not its value) and deviations (knowledge of an error value but not its location). Taking erasures and deviations into account (when they occur) strictly increases the error correction capability of a code: if $\mu$ erasures and $\delta$ deviations occur, then errors of rank $t$ can always be corrected provided that $2t \leq d - 1 + \mu + \delta$, where $d$ is the minimum rank distance of the code. For Gabidulin codes, an important family of maximum rank distance codes, an efficient decoding algorithm is proposed that can properly exploit erasures and deviations. In a network coding application where $n$ packets of length $M$ over $F_q$ are transmitted, the complexity of the decoding algorithm is given by $O(dM)$ operations in an extension field $F_{q^n}$.