Searcharxiv⌕ Search

arXiv subjects

Zhiyuan Yan

Publications and source records attributed to Zhiyuan Yan.

At least 109 records · Page 6Linked to original sources
arXiv2010

Prime Factor Cyclotomic Fourier Transforms with Reduced Complexity over Finite Fields

Discrete Fourier transforms~(DFTs) over finite fields have widespread applications in error correction coding. Hence, reducing the computational complexities of DFTs is of great significance, especially for long DFTs as increasingly longer error control codes are chosen for digital communication and storage systems. Since DFTs involve both multiplications and additions over finite fields and multiplications are much more complex than additions, recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexity. Unfortunately, they have very high additive complexity. Techniques such as common subexpression elimination (CSE) can be used to reduce the additive complexity of CFFTs, but their effectiveness for long DFTs is limited by their complexity. In this paper, we propose prime factor cyclotomic Fourier transforms (PFCFTs), which use CFFTs as sub-DFTs via the prime factor algorithm. When the length of DFTs is prime, our PFCFTs reduce to CFFTs. When the length has co-prime factors, since the sub-DFTs have much shorter lengths, this allows us to use CSE to significantly reduce their additive complexity. In comparison to previously proposed fast Fourier transforms, our PFCFTs achieve reduced overall complexity when the length of DFTs is at least 255, and the improvement significantly increases as the length grows. This approach also enables us to propose efficient DFTs with very long length (e.g., 4095-point), first efficient DFTs of such lengths in the literature. Finally, our PFCFTs are also advantageous for hardware implementation due to their regular structure.

cs.IT↗
arXiv2010

On the Decoder Error Probability of Rank Metric Codes and Constant-Dimension Codes

Rank metric codes and constant-dimension codes (CDCs) have been considered for error control in random network coding. Since decoder errors are more detrimental to system performance than decoder failures, in this paper we investigate the decoder error probability (DEP) of bounded distance decoders (BDDs) for rank metric codes and CDCs. For rank metric codes, we consider a channel motivated by network coding, where errors with the same row space are equiprobable. Over such channels, we establish upper bounds on the DEPs of BDDs, determine the exact DEP of BDDs for maximum rank distance (MRD) codes, and show that MRD codes have the greatest DEPs up to a scalar. To evaluate the DEPs of BDDs for CDCs, we first establish some fundamental geometric properties of the projective space. Using these geometric properties, we then consider BDDs in both subspace and injection metrics and derive analytical expressions of their DEPs for CDCs, over a symmetric operator channel, as functions of their distance distributions. Finally, we focus on CDCs obtained by lifting rank metric codes and establish two important results: First, we derive asymptotically tight upper bounds on the DEPs of BDDs in both metrics; Second, we show that the DEPs for KK codes are the greatest up to a scalar among all CDCs obtained by lifting rank metric codes.

cs.IT↗
arXiv2010

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

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.

cs.IT↗
arXiv2010

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

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.

cs.IT↗
arXiv2009

Bounds on Covering Codes with the Rank Metric

In this paper, we investigate geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an upper bound on the volume of the union of multiple balls with rank radii. Using these geometrical properties, we derive both upper and lower bounds on the minimum cardinality of a code with a given rank covering radius. The geometrical properties and bounds proposed in this paper are significant to the design, decoding, and performance analysis of rank metric codes.

cs.IT↗
arXiv2009

Modified Euclidean Algorithms for Decoding Reed-Solomon Codes

The extended Euclidean algorithm (EEA) for polynomial greatest common divisors is commonly used in solving the key equation in the decoding of Reed-Solomon (RS) codes, and more generally in BCH decoding. For this particular application, the iterations in the EEA are stopped when the degree of the remainder polynomial falls below a threshold. While determining the degree of a polynomial is a simple task for human beings, hardware implementation of this stopping rule is more complicated. This paper describes a modified version of the EEA that is specifically adapted to the RS decoding problem. This modified algorithm requires no degree computation or comparison to a threshold, and it uses a fixed number of iterations. Another advantage of this modified version is in its application to the errors-and-erasures decoding problem for RS codes where significant hardware savings can be achieved via seamless computation.

cs.IT↗
arXiv2009

Construction and Covering Properties of Constant-Dimension Codes

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.

cs.IT↗
arXiv2008

Reduced-Complexity Reed--Solomon Decoders Based on Cyclotomic FFTs

In this paper, we reduce the computational complexities of partial and dual partial cyclotomic FFTs (CFFTs), which are discrete Fourier transforms where spectral and temporal components are constrained, based on their properties as well as a common subexpression elimination algorithm. Our partial CFFTs achieve smaller computational complexities than previously proposed partial CFFTs. Utilizing our CFFTs in both transform- and time-domain Reed--Solomon decoders, we achieve significant complexity reductions.

cs.IT↗
arXiv2008

Cyclotomic FFTs with Reduced Additive Complexities Based on a Novel Common Subexpression Elimination Algorithm

In this paper, we first propose a novel common subexpression elimination (CSE) algorithm for matrix-vector multiplications over characteristic-2 fields. As opposed to previously proposed CSE algorithms, which usually focus on complexity savings due to recurrences of subexpressions, our CSE algorithm achieves two types of complexity reductions, differential savings and recurrence savings, by taking advantage of the cancelation property of characteristic-2 fields. Using our CSE algorithm, we reduce the additive complexities of cyclotomic fast Fourier transforms (CFFTs). Using a weighted sum of the numbers of multiplications and additions as a metric, our CFFTs achieve smaller total complexities than previously proposed CFFTs and other FFTs, requiring both fewer multiplications and fewer additions in many cases.

cs.IT↗
arXiv2008

Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes

For the majority of the applications of Reed-Solomon (RS) codes, hard decision decoding is based on syndromes. Recently, there has been renewed interest in decoding RS codes without using syndromes. In this paper, we investigate the complexity of syndromeless decoding for RS codes, and compare it to that of syndrome-based decoding. Aiming to provide guidelines to practical applications, our complexity analysis differs in several aspects from existing asymptotic complexity analysis, which is typically based on multiplicative fast Fourier transform (FFT) techniques and is usually in big O notation. First, we focus on RS codes over characteristic-2 fields, over which some multiplicative FFT techniques are not applicable. Secondly, due to moderate block lengths of RS codes in practice, our analysis is complete since all terms in the complexities are accounted for. Finally, in addition to fast implementation using additive FFT techniques, we also consider direct implementation, which is still relevant for RS codes with moderate lengths. Comparing the complexities of both syndromeless and syndrome-based decoding algorithms based on direct and fast implementations, we show that syndromeless decoding algorithms have higher complexities than syndrome-based ones for high rate RS codes regardless of the implementation. Both errors-only and errors-and-erasures decoding are considered in this paper. We also derive tighter bounds on the complexities of fast polynomial multiplications based on Cantor's approach and the fast extended Euclidean algorithm.

cs.IT↗
arXiv2008

Constant-Rank Codes

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random network coding. In this paper, we show that constant-rank codes are closely related to constant-dimension codes and we study the properties of constant-rank codes. We first introduce a relation between vectors in $\mathrm{GF}(q^m)^n$ and subspaces of $\mathrm{GF}(q)^m$ or $\mathrm{GF}(q)^n$, and use it to establish a relation between constant-rank codes and constant-dimension codes. We then derive bounds on the maximum cardinality of constant-rank codes with given rank weight and minimum rank distance. Finally, we investigate the asymptotic behavior of the maximal cardinality of constant-rank codes with given rank weight and minimum rank distance.

cs.IT↗
arXiv2008

Packing and Covering Properties of Rank Metric Codes

This paper investigates packing and covering properties of codes with the rank metric. First, we investigate packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their parameters, and investigate their asymptotic covering properties.

cs.IT↗
arXiv2008

On the Decoder Error Probability of Bounded Rank-Distance Decoders for Maximum Rank Distance Codes

In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes that parallel those of maximum distance separable codes. Using these properties, we show that, for MRD codes with error correction capability t, the decoder error probability of bounded rank distance decoders decreases exponentially with t^2 based on the assumption that all errors with the same rank are equally likely.

cs.IT↗
arXiv2007

MacWilliams Identity for Codes with the Rank Metric

The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes. In this paper, we derive the MacWilliams identity for linear codes with the rank metric, and our identity has a different form than that by Delsarte. Using our MacWilliams identity, we also derive related identities for rank metric codes. These identities parallel the binomial and power moment identities derived for codes with the Hamming metric.

cs.IT↗
arXiv2007

Properties of Rank Metric Codes

This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their parameters, and investigate their asymptotic covering properties. Finally, we establish several identities that relate the rank weight distribution of a linear code to that of its dual code. One of our identities is the counterpart of the MacWilliams identity for the Hamming metric, and it has a different form from the identity by Delsarte.

cs.IT↗
arXiv2007

MacWilliams Identity for the Rank Metric

This paper investigates the relationship between the rank weight distribution of a linear code and that of its dual code. The main result of this paper is that, similar to the MacWilliams identity for the Hamming metric, the rank weight distribution of any linear code can be expressed as an analytical expression of that of its dual code. Remarkably, our new identity has a similar form to the MacWilliams identity for the Hamming metric. Our new identity provides a significant analytical tool to the rank weight distribution analysis of linear codes. We use a linear space based approach in the proof for our new identity, and adapt this approach to provide an alternative proof of the MacWilliams identity for the Hamming metric. Finally, we determine the relationship between moments of the rank distribution of a linear code and those of its dual code, and provide an alternative derivation of the rank weight distribution of maximum rank distance codes.

cs.IT↗
arXiv2006

Decoder Error Probability of MRD Codes

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 t^2 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.

cs.IT↗