Bounds for the Multicovering Radii of Reed-Muller Codes with Applications to Stream Ciphers
Designs, Codes and Cryptography
Handbook of Coding Theory
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Johnson type bounds on constant dimension codes
Designs, Codes and Cryptography
Bounds on covering codes with the rank metric
IEEE Communications Letters
Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams
IEEE Transactions on Information Theory
On metrics for error correction in network coding
IEEE Transactions on Information Theory
Network Coding: An Introduction
Network Coding: An Introduction
Recursive code construction for random networks
IEEE Transactions on Information Theory
A Random Linear Network Coding Approach to Multicast
IEEE Transactions on Information Theory
On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum RankDistance Codes
IEEE Transactions on Information Theory
Coding for Errors and Erasures in Random Network Coding
IEEE Transactions on Information Theory
Packing and Covering Properties of Rank Metric Codes
IEEE Transactions on Information Theory
A Rank-Metric Approach to Error Control in Random Network Coding
IEEE Transactions on Information Theory
| Hi-index | 754.84 |
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.