Constant-Rank Codes and Their Connection to Constant-Dimension Codes

Gadouleau, Maximilien; Yan, Zhiyuan

Abstract

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.

Citation

Gadouleau, M., & Yan, Z. (2010). Constant-Rank Codes and Their Connection to Constant-Dimension Codes. IEEE Transactions on Information Theory, 56(7), 3207-3216. https://doi.org/10.1109/tit.2010.2048447

• Journal Article Type: Article
• Publication Date: 2010-07
• Deposit Date: Jan 13, 2012
• Journal: IEEE Transactions on Information Theory
• Print ISSN: 0018-9448
• Electronic ISSN: 1557-9654
• Publisher: Institute of Electrical and Electronics Engineers
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 7
• Pages: 3207-3216
• DOI: https://doi.org/10.1109/tit.2010.2048447
• Public URL: https://durham-repository.worktribe.com/output/1531346