The new construction of rank codes

2008 5th IEEE Annual Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks Workshops

2008

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random 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 constantrank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over sufficiently large extension fields. 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 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.

Section: Asymptotic Resultsmentioning

confidence: 99%

Section: Asymptotic Resultsmentioning

confidence: 99%

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

2008 5th IEEE Annual Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks Workshops

2008

“…For n ≤ m and d R ≤ n, generalized Gabidulin codes [13] can be constructed. For n > m and d R ≤ m, an MRD code can be constructed by transposing a generalized Gabidulin code of length m 1 The Singleton bound in [6] has a different form since array codes are defined over base fields.…”

Section: Packing Properties Of Rank Metric Codesmentioning

confidence: 99%

“…The majority [2], [5]- [7], [12], [13], [15], [17], [18] of previous works focus on rank distance properties, code construction, and efficient decoding of rank metric codes. Some previous works focus on the packing and covering properties of rank metric codes.…”

Section: Introductionmentioning

confidence: 99%

Packing and Covering Properties of Rank Metric Codes

IEEE Trans. Inform. Theory

2008

“…An explicit construction of MRD codes (these codes are referred to as Gabidulin codes) was also given in [1], and this construction was extended in [6]. Also, a decoding algorithm that parallels the extended Euclidean algorithm (EEA) was proposed for MRD codes.…”

Section: Introductionmentioning

confidence: 99%

Decoder Error Probability of MRD Codes

2006 IEEE Information Theory Workshop - ITW '06 Chengdu

2006

In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. Using this new concept, we derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable (MDS) codes. Using these properties, we show that the decoder error probability of MRD codes with error correction capability t decreases exponentially with t2 based on the assumption that all errors with the same rank are equally likely. We argue that the channel based on this assumption is an approximation of a channel corrupted by crisscross errors. I. INTRODUCTIONThe Hamming metric has often been considered the most relevant metric for error-control codes so far. Recently, the rank metric [1] has attracted some attention due to its relevance to space-time coding [2] and storage applications [3]. In [4], space-time block codes with good rank properties have been proposed. Rank metric codes are used to correct crisscross errors that can be found in memory chip arrays and magnetic tapes [3]. Rank metric codes have been used in public-key cryptosystems as well [5].In [1], a Singleton bound on the minimum rank distance of rank metric codes was established, and codes that attain this bound were called maximum rank distance (MRD) codes. An explicit construction of MRD codes (these codes are referred to as Gabidulin codes) was also given in [1], and this construction was extended in [6]. Also, a decoding algorithm that parallels the extended Euclidean algorithm (EEA) was proposed for MRD codes.In this paper, we investigate the performance of MRD codes when used to protect data from additive errors based on two assumptions. First, we assume all errors with the same rank are equally likely. We argue that the channel based on this assumption is an approximation of a channel corrupted by crisscross errors (see Section IV for details). Second, we assume that a bounded rank distance decoder is used, with error correction capability t. If the error has rank no more than t, the decoder gives the correct codeword. When the error