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Efficient Bounded Distance Decoders for BarnesWall Lattices
, 2008
"... We describe a new family of parallelizable bounded distance decoding algorithms for the BarnesWall lattices, and analyze their decoding complexity. The algorithms are parameterized by the number p = 4k ≤ N² of available processors, work for BarnesWall lattices in arbitrary dimension N = 2n, correc ..."
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We describe a new family of parallelizable bounded distance decoding algorithms for the BarnesWall lattices, and analyze their decoding complexity. The algorithms are parameterized by the number p = 4k ≤ N² of available processors, work for BarnesWall lattices in arbitrary dimension N = 2n, correct any error up to squared unique decoding radius d² min /4, and run in worstcase time O(N log² N / √ p). Depending on the value of the parameter p, this yields efficient decoding algorithms ranging from a fast sequential algorithm with quasilinear decoding complexity O(N log² N), to a fully parallel decoding circuit with polylogarithmic depth O(log² N) and polynomially many arithmetic gates.
Construction of BarnesWall Lattices from Linear Codes over Rings
, 2012
"... Dense lattice packings can be obtained via the wellknown Construction A from binary linear codes. In this paper, we use an extension of Construction A called Construction A ′ to obtain BarnesWall lattices from linear codes over polynomials rings. To obtain the BarnesWall lattice BW2m in C2m for a ..."
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Dense lattice packings can be obtained via the wellknown Construction A from binary linear codes. In this paper, we use an extension of Construction A called Construction A ′ to obtain BarnesWall lattices from linear codes over polynomials rings. To obtain the BarnesWall lattice BW2m in C2m for any m ≥ 1, we first identify a linear code C2m over the quotient ring Um = F2[u]�u m and then propose a mapping ψ: Um → Z[i] such that the code L2m = ψ(C2m) is a lattice constellation. Further, we show that L2m has the cubic shaping property when m is even. Finally, we show that BW2m can be obtained through Construction A ′ as BW2m =(1+i)mZ[i] 2m
Practical encoders and decoders for Euclidean codes from BarnesWall lattices
 IEEE Trans. Commun. Available
"... lattice codes for communication over additive white Gaussian noise (AWGN) channels. We introduce Construction A ′ of complex BW lattices that makes new connection between linear codes over polynomial rings and lattices. We show that Construction A ′ of BW lattices is equivalent to the multilevel co ..."
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lattice codes for communication over additive white Gaussian noise (AWGN) channels. We introduce Construction A ′ of complex BW lattices that makes new connection between linear codes over polynomial rings and lattices. We show that Construction A ′ of BW lattices is equivalent to the multilevel construction from ReedMuller codes proposed by Forney. To decode the BW lattice code, we adapt the lowcomplexity sequential BW lattice decoder (SBWD) proposed by Micciancio and Nicolosi. First we study the error performance of SBWD for decoding the infinite lattice, and demonstrate that it is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise trimming technique, where the received vector is appropriately scaled before applying the SBWD. We show that the noise trimming technique is most effective for decoding BW lattice codes in smaller dimensions, while the gain diminishes for decoding codes in larger dimensions. Index Terms—BarnesWall lattices, lattice codes, lowcomplexity lattice decoders. I.
List Decoding BarnesWall Lattices
, 2012
"... The question of list decoding errorcorrecting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in R N, and their many shared applications across complexity theory, crypt ..."
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The question of list decoding errorcorrecting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in R N, and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L ⊆ R N, given a target vector r ∈ R N and a distance parameter d, output the set of all lattice points w ∈ L that are within distance d of r. In this work we focus on combinatorial and algorithmic questions related to list decoding for the wellstudied family of BarnesWall lattices. Our main contributions are twofold: 1. We give tight (up to polynomials) combinatorial bounds on the worstcase list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattice’s minimum distance (in the Euclidean norm). 2. Building on the unique decoding algorithm of Micciancio and Nicolosi (ISIT ’08), we give a listdecoding algorithm that runs in time polynomial in the lattice dimension and worstcase list size, for any error radius. Moreover, our algorithm is highly parallelizable, and with sufficiently many processors can run in parallel time only polylogarithmic in the lattice dimension. In particular, our results imply a polynomialtime listdecoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural errorcorrecting codes posed by the Johnson radius.