An improved LLL algorithm

Luk, Franklin T.; Tracy, Daniel M.

doi:10.1016/j.laa.2007.02.029

Cited by 19 publications

(12 citation statements)

References 4 publications

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“…The LLL algorithm [12], [13], [15] is widely used in lattice reduction aided decoding, because it practically produces reasonably good results with low complexity. A matrix form of the LLL algorithm can be found in [14].…”

Section: Resultsmentioning

confidence: 99%

A Jacobi Method for Lattice Basis Reduction

Qiao

2012

2012 Spring Congress on Engineering and Technology

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Abstract-Lattice reduction aided decoding has been successfully used in wireless communications. In this paper, we propose a Jacobi method for lattice basis reduction. Jacobi method is attractive, because it is inherently parallel. Thus high performance can be achieved by exploiting multiprocessor and/or multicore architectures. We also present our experimental results on the convergence of our method and the comparison with the LLL algorithm, a lattice basis reduction method widely used in wireless communication applications.

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Section: Resultsmentioning

confidence: 99%

A Jacobi Method for Lattice Basis Reduction

Qiao

2012

2012 Spring Congress on Engineering and Technology

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“…Then from (10), (27) Thus, for , we have (28) which implies that the initial radius of the modified SE enumeration called in each iteration of Algorithm M-RED-1 is bounded above by (29) Combining (20) and (29), the overall complexity of the enumeration part of M-RED-1 is given by (30) Now, we consider the complexity of the gcd conditions checking part. It is easy to verify that during the process of each SE enumeration, the number of lattice points for which the gcd conditions are checked is proportional to the volume of the -dim sphere of radius (29). On the other hand, from the analysis in Section III-B, the elementary operations performed by the gcd condition checking per lattice point is .…”

Section: Complexity Analysismentioning

confidence: 99%

“…[18], [29], [34], [35], [42], [43] adopt the QR decomposition approach, since the QR decomposition can be performed efficiently and numerically more stable than GSO. …”

Section: ) Lattices and Basesmentioning

confidence: 99%

HKZ and Minkowski Reduction Algorithms for Lattice-Reduction-Aided MIMO Detection

Zhang

Qiao

Wei

2012

IEEE Trans. Signal Process.

102

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Abstract-Recently, lattice reduction has been widely used for signal detection in multiinput multioutput (MIMO) communications. In this paper, we present three novel lattice reduction algorithms. First, using a unimodular transformation, a significant improvement on an existing Hermite-Korkine-Zolotareff-reduction algorithm is proposed. Then, we present two practical algorithms for constructing Minkowski-reduced bases. To assess the output quality, we compare the orthogonality defect of the reduced bases produced by LLL algorithm and our new algorithms, and find that in practice Minkowski-reduced basis vectors are the closest to being orthogonal. An error-rate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented. To this aim, the proximity factor is employed as a measurement. We improve some existing results and derive upper bounds for the proximity factors of Minkowski-reduction-aided decoding (MRAD) to show that MRAD can achieve the same diversity order with infinite lattice decoding (ILD).

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“…It was pointed out in Section 3 that the LLL algorithm can improve conditioning by transforming a given matrix into a reduced one via imposing conditions (1) and (2). In this section, we propose a pivoting strategy into the LLL algorithm that may further reduce a reduced matrix.…”

Section: Further Conditioningmentioning

confidence: 99%

“…Therefore, the procedure Decrease in the LLL algorithm can improve the conditioning of a triangular matrix by enforcing Condition (1). Condition (2) requires that the diagonal elements be in loosely increasing order from top to bottom. The smaller the ω is, the more loosely the diagonal is ordered.…”

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confidence: 99%