arXiv subjects

Brian M. Kurkoski

Publications and source records attributed to Brian M. Kurkoski.

28 records · Page 2Linked to original sources

arXiv2018

Construction D$^\prime$ Lattices from Quasi-Cyclic Low-Density Parity-Check Codes

Recently, Branco da Silva and Silva described an efficient encoding and decoding algorithm for Construction D$^\prime$ lattices. Using their algorithm, we propose a Construction D$^\prime$ lattice based on binary quasi-cyclic low-density parity-check (QC-LPDC) codes and single parity-check product codes. The underlying codes designed by the balanced-distances rule contribute in a balanced manner to the squared minimum distance of the constructed lattice, which results in a high lattice coding gain. The proposed lattice based on IEEE 802.16e QC-LDPC codes is shown to provide competitive error-rate performance on the power-unconstrained additive white Gaussian noise channel.

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arXiv2017

Proceedings of Workshop AEW10: Concepts in Information Theory and Communications

The 10th Asia-Europe workshop in "Concepts in Information Theory and Communications" AEW10 was held in Boppard, Germany on June 21-23, 2017. It is based on a longstanding cooperation between Asian and European scientists. The first workshop was held in Eindhoven, the Netherlands in 1989. The idea of the workshop is threefold: 1) to improve the communication between the scientist in the different parts of the world; 2) to exchange knowledge and ideas; and 3) to pay a tribute to a well respected and special scientist.

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arXiv2016

Low-Dimensional Shaping for High-Dimensional Lattice Codes

We propose two low-complexity lattice code constructions that have competitive coding and shaping gains. The first construction, named systematic Voronoi shaping, maps short blocks of integers to the dithered Voronoi integers, which are dithered integers that are uniformly distributed over the Voronoi region of a low-dimensional shaping lattice. Then, these dithered Voronoi integers are encoded using a high-dimensional lattice retaining the same shaping and coding gains of low and high-dimensional lattices. A drawback to this construction is that there is no isomorphism between the underlying message and the lattice code, preventing its use in applications such as compute-and- forward. Therefore we propose a second construction, called mixed nested lattice codes, in which a high-dimensional coding lattice is nested inside a concatenation of low-dimensional shaping lattices. This construction not only retains the same shaping/coding gains as first construction but also provides the desired algebraic structure. We numerically study these methods, for point-to-point channels as well as compute-and-forward using low-density lattice codes (LDLCs) as coding lattices and E8 and Barnes-Wall as shaping lattices. Numerical results indicate a shaping gain of up to 0.86 dB, compared to the state-of-the-art of 0.4 dB; furthermore, the proposed method has lower complexity than state-of-the-art approaches.

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arXiv2016

Encoding and Indexing of Lattice Codes

Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $Λ_{\textrm{c}}$ and shaping lattice $Λ_{\textrm{s}}$ satisfy $Λ_{\textrm{s}} \subseteq Λ_{\textrm{c}}$, then $Λ_{\textrm{c}} / Λ_{\textrm{s}}$ is a quotient group that can be used to form a (nested) lattice code $\mathcal{C}$. Conway and Sloane's method of encoding and indexing does not apply when the lattices are not self-similar. Results are provided for two classes of lattices. (1) If $Λ_{\textrm{c}}$ and $Λ_{\textrm{s}}$ both have generator matrices in triangular form, then encoding is always possible. (2) When $Λ_{\textrm{c}}$ and $Λ_{\textrm{s}}$ are described by full generator matrices, if a solution to a linear diophantine equation exists, then encoding is possible. In addition, special cases where $\mathcal{C}$ is a cyclic code are also considered. A condition for the existence of a group homomorphism between the information and $\mathcal{C}$ is given. The results are applicable to a variety of coding lattices, including Construction A, Construction D and LDLCs. The $D_4$, $E_8$ and convolutional code lattices are shown to be good choices for the shaping lattice. Thus, a lattice code $\mathcal{C}$ can be designed by selecting $Λ_{\textrm{c}}$ and $Λ_{\textrm{s}}$ separately, avoiding competing design requirements of self-similar lattice codes.

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arXiv2014

Quantization of Binary-Input Discrete Memoryless Channels

The quantization of the output of a binary-input discrete memoryless channel to a smaller number of levels is considered. An algorithm which finds an optimal quantizer, in the sense of maximizing mutual information between the channel input and the quantizer output is given. This result holds for arbitrary channels, in contrast to previous results for restricted channels or a restricted number of quantizer outputs. In the worst case, the algorithm complexity is cubic $M^3$ in the number of channel outputs $M$. Optimality is proved using the theorem of Burshtein, Della Pietra, Kanevsky, and Nádas for mappings which minimize average impurity for classification and regression trees.

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arXiv2011

Threshold Improvement of Low-Density Lattice Codes via Spatial Coupling

Spatially-coupled low-density lattice codes (LDLC) are constructed using protographs. Using Monte Carlo density evolution using single-Gaussian messages, we observe that the threshold of the spatially-coupled LDLC is within 0.22 dB of capacity of the unconstrained power channel. This is in contrast with a 0.5 dB noise threshold for the conventional LDLC lattice construction.

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arXiv2011

The E8 Lattice and Error Correction in Multi-Level Flash Memory

A construction using the E8 lattice and Reed-Solomon codes for error-correction in flash memory is given. Since E8 lattice decoding errors are bursty, a Reed-Solomon code over GF($2^8$) is well suited. This is a type of coded modulation, where the Euclidean distance of the lattice, which is an eight-dimensional signal constellation, is combined with the Hamming distance of the code. This system is compared with the conventional technique for flash memories, BCH codes using Gray-coded PAM. The described construction has a performance advantage of 1.6 to 1.8 dB at a probability of word error of $10^{-6}$. Evaluation is at high data rates of 2.9 bits/cell for flash memory cells that have an uncoded data density of 3 bits/cell.

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arXiv2010

Rewritable Codes for Flash Memories Based Upon Lattices, and an Example Using the E8 Lattice

A rewriting code construction for flash memories based upon lattices is described. The values stored in flash cells correspond to lattice points. This construction encodes information to lattice points in such a way that data can be written to the memory multiple times without decreasing the cell values. The construction partitions the flash memory's cubic signal space into blocks. The minimum number of writes is shown to be linear in one of the code parameters. An example using the E8 lattice is given, with numerical results.

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arXiv2009

Belief-Propagation Decoding of Lattices Using Gaussian Mixtures

A belief-propagation decoder for low-density lattice codes is given which represents messages explicitly as a mixture of Gaussians functions. The key component is an algorithm for approximating a mixture of several Gaussians with another mixture with a smaller number of Gaussians. This Gaussian mixture reduction algorithm iteratively reduces the number of Gaussians by minimizing the distance between the original mixture and an approximation with one fewer Gaussians. Error rates and noise thresholds of this decoder are compared with those for the previously-proposed decoder which discretely quantizes the messages. The error rates are indistinguishable for dimension 1000 and 10000 lattices, and the Gaussian-mixture decoder has a 0.2 dB loss for dimension 100 lattices. The Gaussian-mixture decoder has a loss of about 0.03 dB in the noise threshold, which is evaluated via Monte Carlo density evolution. Further, the Gaussian-mixture decoder uses far less storage for the messages.

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arXiv2008

Message-Passing Decoding of Lattices Using Gaussian Mixtures

A lattice decoder which represents messages explicitly as a mixture of Gaussians functions is given. In order to prevent the number of functions in a mixture from growing as the decoder iterations progress, a method for replacing N Gaussian functions with M Gaussian functions, with M < N, is given. A squared distance metric is used to select functions for combining. A pair of selected Gaussians is replaced by a single Gaussian with the same first and second moments. The metric can be computed efficiently, and at the same time, the proposed algorithm empirically gives good results, for example, a dimension 100 lattice has a loss of 0.2 dB in signal-to-noise ratio at a probability of symbol error of 10^{-5}.

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