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

Zhang, Wen; Qiao, Sanzheng; Wei, Yimin

doi:10.1109/tsp.2012.2210708

Cited by 44 publications

(103 citation statements)

References 54 publications

(150 reference statements)

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“…. Lines 5 to 7 are designed to plug new vectors found into the lattice basis, and the basis expansion method in [10] can do this efficiently. Other basis expansion methods include using LLL reduction [30] or employing the Hermite normal form of the coefficient matrix [1, Lem.…”

Section: B Algorithm Descriptionmentioning

confidence: 99%

“…Other basis expansion methods include using LLL reduction [30] or employing the Hermite normal form of the coefficient matrix [1, Lem. 7.1], but both of them have higher complexity than the one in [10]. Lines 8 to 10 restore the upper triangular property of R, and these be alternatively implemented by performing another QR decomposition.…”

Section: B Algorithm Descriptionmentioning

confidence: 99%

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Boosted KZ and LLL Algorithms

Lyu

Ling

2017

IEEE Trans. Signal Process.

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Abstract-There exist two issues among popular lattice reduction (LR) algorithms that should cause our concern. The first one is Korkine-Zolotarev (KZ) and Lenstra-Lenstra-Lovász (LLL) algorithms may increase the lengths of basis vectors. The other is KZ reduction suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, despite its superior theoretical upper bounds. To address these limitations, we improve the size reduction steps in KZ and LLL to set up two new efficient algorithms, referred to as boosted KZ and LLL, for solving the shortest basis problem (SBP) with exponential and polynomial complexity, respectively. Both of them offer better actual performance than their classic counterparts, and the performance bounds for KZ are also improved. We apply them to designing integer-forcing (IF) linear receivers for multiinput multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages.

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Section: B Algorithm Descriptionmentioning

confidence: 99%

Section: B Algorithm Descriptionmentioning

confidence: 99%

Boosted KZ and LLL Algorithms

Lyu

Ling

2017

IEEE Trans. Signal Process.

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“…3) Comparison with Minkowski reduction. Recall that a lattice basis B is called Minkowski reduced if for any integers c 1 , ..., c n such that c i , ..., c n are altogether coprime, it has b 1 c 1 + · · · + b n c n ≥ b i for 1 ≤ i ≤ n [15]. For a Minkowski reduced basis, it satisfies [15]…”

Section: B Discussionmentioning

confidence: 99%

On Low-Complexity Lattice Reduction Algorithms for Large-Scale MIMO Detection: The Blessing of Sequential Reduction

Lyu¹,

Wen²,

Weng³

et al. 2020

IEEE Trans. Signal Process.

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Lattice reduction is a popular preprocessing strategy in multiple-input multiple-output (MIMO) detection. In a quest for developing a low-complexity reduction algorithm for largescale problems, this paper investigates a new framework called sequential reduction (SR), which aims to reduce the lengths of all basis vectors. The performance upper bounds of the strongest reduction in SR are given when the lattice dimension is no larger than 4. The proposed new framework enables the implementation of a hash-based low-complexity lattice reduction algorithm, which becomes especially tempting when applied to large-scale MIMO detection. Simulation results show that, compared to other reduction algorithms, the hash-based SR algorithm exhibits the lowest complexity while maintaining comparable error performance.

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“…It is also possible to extend the algebraic LLL to algebraic KZ/Minkowski reduction [26]. Although designing an algorithm for the shortest vector problem (SVP) over the algebraic lattices may encounter the difficulty of enumerating points in a depth-first enumeration algorithm, we can alternatively use SVP on real lattices.…”

Section: Beyond Algebraic Lllmentioning

confidence: 99%

“…For instance, only 1/4 of the points within a Euclidean ball need to be enumerated in Gaussian integers as |Z [i] × | = 4 (as used in [27]), and only 1/6 of the points need to be enumerated in Eisenstein integers as |Z [ω] × | = 6. Although KZ/Minkowski reduction algorithms do not have the constraint due to Lovasz's condition, their basis expansion process [26] still requires the rings to be Euclidean.…”

Section: Beyond Algebraic Lllmentioning

confidence: 99%

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Lyu

Porter

Ling

2018

2018 IEEE Information Theory Workshop (ITW)

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Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices.In this work, we study such lattices and their reduction algorithms. First, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice in polynomial time if the chosen ring is Euclidean. Second, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, as an application, we use the algebraic algorithms to find the network coding matrices in compute-and-forward.Such lattice reduction-based approaches have low complexity which is not dictated by the signal-to-noise (SNR) ratio. Moreover, such approaches can not only preserve the degree-of-freedom of computation rates, but ensure the independence in the code space as well. Index Termslattice reduction, Gauss's algorithm, LLL, compute-and-forward. Initially coding lattices from ConstructionA over rational integers Z or Gaussian integers Z[i] are the main enabler in showing the achievable information rate in C&F. Recently, however, the Construction A lattices have been extended to over the ring of Eisenstein integers Z[ω] [4], [5] and other rings of imaginary quadratic integers [6], This work was presented in part at the IEEE Information Theory Workshop 2018, Guangzhou, China, and submitted in part to the IEEE Information Theory Workshop 2019, Visby, Gotland, Sweden. S. Lyu is with the College

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

Cited by 44 publications

References 54 publications

Boosted KZ and LLL Algorithms

Boosted KZ and LLL Algorithms

On Low-Complexity Lattice Reduction Algorithms for Large-Scale MIMO Detection: The Blessing of Sequential Reduction

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

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