Packing and Covering Properties of Subspace Codes for Error Control in Random Linear Network Coding

TL;DR

This paper explores the geometric properties and bounds of subspace codes and constant-dimension codes for error correction in network coding, comparing two metrics and their implications for code design.

Contribution

It derives bounds and asymptotic rates for packing and covering subspace codes under two metrics, revealing geometric differences and optimality conditions.

Findings

Optimal packing CDCs are asymptotically optimal for subspace codes when dimension is half the length.

CDC-based codes suffer limited rate loss compared to general subspace codes.

Optimal covering CDCs can construct asymptotically optimal covering codes with the injection metric.

Abstract

Codes in the projective space and codes in the Grassmannian over a finite field - referred to as subspace codes and constant-dimension codes (CDCs), respectively - have been proposed for error control in random linear network coding. For subspace codes and CDCs, a subspace metric was introduced to correct both errors and erasures, and an injection metric was proposed to correct adversarial errors. In this paper, we investigate the packing and covering properties of subspace codes with both metrics. We first determine some fundamental geometric properties of the projective space with both metrics. Using these properties, we then derive bounds on the cardinalities of packing and covering subspace codes, and determine the asymptotic rates of optimal packing and optimal covering subspace codes with both metrics. Our results not only provide guiding principles for the code design for error control in random linear network coding, but also illustrate the difference between the two metrics from a geometric perspective. In particular, our results show that optimal packing CDCs are optimal packing subspace codes up to a scalar for both metrics if and only if their dimension is half of their length (up to rounding). In this case, CDCs suffer from only limited rate loss as opposed to subspace codes with the same minimum distance. We also show that optimal covering CDCs can be used to construct asymptotically optimal covering subspace codes with the injection metric only.