Construction and Covering Properties of Constant-Dimension Codes

TL;DR

This paper introduces augmented KK codes for constant-dimension codes, providing improved constructions and decoding algorithms, and investigates their covering properties to optimize error correction in network coding.

Contribution

It presents a new class of CDCs called augmented KK codes with higher cardinalities and a low-complexity decoding algorithm, advancing the design of error-correcting codes for network coding.

Findings

Augmented KK codes have larger cardinalities than previous CDCs.

The proposed decoding algorithm corrects more errors than bounded subspace distance decoders.

Liftings of rank metric codes have the highest possible covering radius, indicating they are not optimal for packing.

Abstract

Constant-dimension codes (CDCs) have been investigated for noncoherent error correction in random network coding. The maximum cardinality of CDCs with given minimum distance and how to construct optimal CDCs are both open problems, although CDCs obtained by lifting Gabidulin codes, referred to as KK codes, are nearly optimal. In this paper, we first construct a new class of CDCs based on KK codes, referred to as augmented KK codes, whose cardinalities are greater than previously proposed CDCs. We then propose a low-complexity decoding algorithm for our augmented KK codes using that for KK codes. Our decoding algorithm corrects more errors than a bounded subspace distance decoder by taking advantage of the structure of our augmented KK codes. In the rest of the paper we investigate the covering properties of CDCs. We first derive bounds on the minimum cardinality of a CDC with a given covering radius and then determine the asymptotic behavior of this quantity. Moreover, we show that liftings of rank metric codes have the highest possible covering radius, and hence liftings of rank metric codes are not optimal packing CDCs. Finally, we construct good covering CDCs by permuting liftings of rank metric codes.