Search

The quality and quantity of data used for training greatly influence the performance and effectiveness of deep learning models. In the context of error correction, it is essential to generate high-quality samples that are neither excessively noisy nor entirely correct but close to the decoding region's decision boundary. To accomplish this objective, this paper utilizes a restricted version of a recent result on Importance Sampling (IS) distribution for fast performance evaluation of linear codes. The IS distribution is used over the segmented observation space and integrated with active learning. This combination allows for the iterative generation of samples from the shells whose acquisition functions, defined as the error probabilities conditioned on each shell, fall within a specific range. By intelligently sampling based on the proposed IS distribution, significant improvements are demonstrated in the performance of BCH(63,36) and BCH(63,45) codes with cycle-reduced parity-check matrices. The proposed IS-based-active Weight Belief Propagation (WBP) decoder shows improvements of up to 0.4dB in the waterfall region and up to 1.9dB in the error-floor region of the BER curve, over the conventional WBP. This approach can be easily adapted to generate efficient samples to train any other deep learning-based decoder.

LDPC codes based on multiple-edge protographs potentially have larger minimum distances compared to their counterparts, single-edge protographs. However, considering different features of their Tanner graph, such as short cycles, girth and other graphical structures, is harder than for Tanner graphs from single-edge protographs. In this paper, we provide a novel approach to construct the parity-check matrix of an LDPC code which is based on trades obtained from block designs. We employ our method to construct two important categories of LDPC codes; quasi-cyclic (QC) LDPC and spatially-coupled LDPC (SC-LDPC) codes. We use those trade-based matrices to define base matrices of multiple-edge protographs. The construction of exponent matrices corresponding to these base matrices has less complexity compared to the ones proposed in the literature. We prove that these base matrices result in QC-LDPC codes with smaller lower bounds on the lifting degree than existing ones. There are three categories of SC-LDPC codes: periodic, time-invariant and time-varying. Constructing the parity-check matrix of the third one is more difficult because of the time dependency in the parity-check matrix. We use a trade-based matrix to obtain the parity-check matrix of a time-varying SC-LDPC code in which each downwards row displacement of the trade-based matrix yields syndrome matrices of a particular time. Combining the different row shifts the whole parity-check matrix is obtained. Our proposed method to construct parity-check and base matrices from trade designs is applicable to any type of super-simple directed block designs. We apply our technique to directed designs with smallest defining sets containing at least half of the blocks. To demonstrate the significance of our contribution, we provide a number of numerical and simulation results.

Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, has been generalized to a lattice construction from totally real and complex multiplication (CM) fields. This generalized algebraic Construction A of lattices provides signal space diversity, intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper, we construct full-diversity algebraic lattices for BF channels using Construction A over totally real number fields. We propose two new decoding methods for these lattices which have complexity that grows linearly in the dimension of the lattice. The first decoder is proposed for generalized Construction A lattices with a binary LDPC code as underlying code. This decoding method contains iterative and non-iterative phases. In order to implement the iterative phase, we propose the definition of a parity-check matrix and Tanner graph for Construction A lattices. We also prove that using an underlying LDPC code that achieves the outage probability limit over one-BF channel, the constructed algebraic LDPC lattices together with the proposed decoding method admit diversity order n. Then, we modify the proposed algorithm by removing its iterative phase which enables full-diversity practical decoding of all generalized Construction A lattices without any assumption about their underlying code. We provide some instances showing that algebraic Construction A lattices obtained from binary codes outperform the ones based on non-binary codes in BF channels. We generalize algebraic Construction A lattices over a wider family of number fields namely monogenic number fields., Comment: This work is accepted for Publication in IEEE Transactions on Information Theory. Part of this work has been presented at ISIT 2016, Spain. arXiv admin note: substantial text overlap with arXiv:1612.04039; text overlap with arXiv:1404.2904 by other authors

A new nonlinear Rao-Nam like symmetric key encryption scheme is presented in this paper. QC-LDPC lattices that are practically implementable in high dimensions due to their low complexity encoding and decoding algorithms, are used in our design. Then, a joint scheme is proposed which is capable of encrypting, encoding and data modulation simultaneously. The proposed cryptosystem withstands all variants of chosen plaintext attacks applied on Rao-Nam like cryptosystems due to its nonlinearity. The sparseness of the parity-check matrix of QC-LDPC lattices, quasi-cyclic nature of their generator and parity-check matrices, simple hardware structure for generating intentional error vector, permutation and nonlinear functions, result in a small key size for our scheme. The lattice codes related to the lattices used in this paper have high rate which are suitable for bandlimited AWGN channels. Therefore, the joint scheme based on these lattices facilitates secure, reliable and efficient data transmission in bandlimited AWGN channels., Comment: 30 pages, 4 figures

Ciphertext-policy hierarchical attribute-based encryption (CP-HABE) is a promising cryptographic primitive for enforcing the fine-grained access control with scalable key delegation and user revocation mechanisms on the outsourced encrypted data in a cloud. Wang et al. (2011) proposed the first CP-HABE scheme and showed that the scheme is semantically secure in the random oracle model [4, 5]. Due to some weakness in its key delegation mechanism, by presenting two attacks, we demonstrate the scheme does not offer any confidentiality and fine-grained access control. In this way, anyone who has just one attribute can recover any outsourced encrypted data in the cloud.

One of the phenomena that causes high decoding failure rates is trapping sets. Characterization of $(a,b)$ elementary trapping sets (ETSs), their graphical properties and the lower bounds on their size in variable regular LDPC codes with column weights 3, 4, 5 and 6, where $a$ is the size of the ETS and $b$ is the number of degree-one check nodes, have been an interesting subject among researchers. Although progressive-edge-growth method (PEG) to construct LDPC codes free of an specific ETS has been proposed in the literature, it is mostly applied to LDPC codes with column weight 3. In this paper, we focus on constructing QC-LDPC codes with column weight 4 whose Tanner graphs are free of small ETSs. Using coloring the edges of the variable node (VN) graph corresponding to an ETS, we provide the sufficient conditions to obtain QC-LDPC codes with column weight 4, girth 6 and free of $(5,b)$ ETSs, where $b\leq4$, and $(6,b)$ ETs, where $b\leq2$. Moreover, for $(4,n)$-regular QC-LDPC codes with girth 8, we present a method to remove $(7,4)$ ETSs from Tanner graphs., Comment: 12

Neural network decoding algorithms are recently introduced by Nachmani et al. to decode high-density parity-check (HDPC) codes. In contrast with iterative decoding algorithms such as sum-product or min-sum algorithms in which the weight of each edge is set to $1$, in the neural network decoding algorithms, the weight of every edge depends on its impact in the transmitted codeword. In this paper, we provide a novel \emph{feed-forward neural network lattice decoding algorithm} suitable to decode lattices constructed based on Construction A, whose underlying codes have HDPC matrices. We first establish the concept of feed-forward neural network for HDPC codes and improve their decoding algorithms compared to Nachmani et al. We then apply our proposed decoder for a Construction A lattice with HDPC underlying code, for which the well-known iterative decoding algorithms show poor performances. The main advantage of our proposed algorithm is that instead of assigning and training weights for all edges, which turns out to be time-consuming especially for high-density parity-check matrices, we concentrate on edges which are present in most of $4$-cycles and removing them gives a girth-$6$ Tanner graph. This approach, by slight modifications using updated LLRs instead of initial ones, simultaneously accelerates the training process and improves the error performance of our proposed decoding algorithm., Comment: To appear in ITW 2018

One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An $(a,b)$ elementary trapping set, or simply an ETS where $a$ is the size and $b$ is the number of degree-one check nodes and $\frac{b}{a}<1$, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected $(3,n)$-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs., Comment: 15, pages, 2 figures

A locally irregular graph is a graph whose adjacent vertices have distinct degrees, a regular graph is a graph where each vertex has the same degree and a locally regular graph is a graph where for every two adjacent vertices u, v, their degrees are equal. In this work, we study the set of all problems which are related to decomposition of graphs into regular, locally regular and/or locally irregular subgraphs and we present some polynomial time algorithms, NP-completeness results, lower bounds and upper bounds for them. Among our results, one of our lower bounds makes use of mutually orthogonal Latin squares which is relatively novel., Comment: 31 pages, 8 figures

A Not-All-Equal (NAE) decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts such that each vertex in $G$ has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph $G$ is a decomposition of the vertices of $G$ into two parts $A$ and $B$ such that each vertex in the graph $G$ has exactly one neighbor in part $A$. Among our results, we show that for a given graph $G$, if $G$ does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether $G$ has a 1-in-Degree decomposition. In sharp contrast, we prove that for every $r$, $r\geq 3$, for a given $r$-regular bipartite graph $G$ determining whether $G$ has a 1-in-Degree decomposition is $ \mathbf{NP} $-complete. These complexity results have been especially useful in proving $ \mathbf{NP} $-completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph $G$ determining whether there is a vector in the null-space of the 0,1-adjacency matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP} $-complete. Among other results, we introduce a new version of {Planar 1-in-3 SAT} and we prove that this version is also $ \mathbf{NP} $-complete. In consequence of this result, we show that for a given planar $(3,4)$-semiregular graph $G$ determining whether there is a vector in the null-space of the 0,1-incidence matrix of $G$ such that its entries belong to $\{\pm 1,\pm 2\}$ is $\mathbf{NP} $-complete., Comment: To appear in Algorithmica

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions., Comment: Submitted for possible publication

In this paper, we define two matrices named as "difference matrices", denoted by $D$ and $DD$ which significantly contribute to achieve regular single-edge QC-LDPC codes with the shortest length and the certain girth as well as regular and irregular multiple-edge QC-LDPC codes. Making use of these matrices, we obtain necessary and sufficient conditions to have single-edge $(m,n)$-regular QC-LDPC codes with girth 6, 10 and 12. Additionally, for girth 6, we achieve all non-isomorphic codes with the minimum lifting degree, $N$, for $m=4$ and $5\leq n\leq 11$, and present an exponent matrix for each minimum distance. For girth 10, we provide a lower bound on the lifting degree which is tighter than the existing bound. More important, for an exponent matrix whose first row and first column are all-zero, we demonstrate that the non-existence of 8-cycles proves the non-existence of 6-cycles related to the first row of the exponent matrix too. All non-isomorphic QC-LDPC codes with girth 10 and $n=5,6$ whose numbers are more than those presented in the literature are provided. For $n=7,8$ we decrease the lifting degrees from 159 and 219 to 145 and 211, repectively. For girth 12, a lower bound on the lifting degree is achieved. For multiple-edge category, for the first time a lower bound on the lifting degree for both regular and irregular QC-LDPC codes with girth 6 is achieved. We also demonstrate that the achieved lower bounds on multiple-edge $(4,n)$-regular QC-LDPC codes with girth 6 are tight and the resultant codes have shorter length compared to their counterparts in single-edge codes. Additionally, difference matrices help to reduce the conditions of considering 6-cycles from seven states to five states. We obtain multiple-edge $(4,n)$-regular QC-LDPC codes with girth 8 and $n=4,6,8$ with the shortest length., Comment: 33 pages

In this paper we give lower bounds on the size of $(a,b)$ elementary trapping sets (ETSs) belonging to variable-regular LDPC codes with any girth, $g$, and irregular ones with girth 8, where $a$ is the size, $b$ is the number of degree-one check nodes and satisfy the inequality $\frac{b}{a}<1$. Our proposed lower bounds are analytical, rather than exhaustive search-based, and based on graph theories. The numerical results in the literarture for $g=6,8$ for variable-regular LDPC codes match our results. Some of our investigations are independent of the girth and rely on the variables $a$, $b$ and $\gamma$, the column weight value, only. We prove that for an ETS belonging to a variable-regular LDPC code with girth 8 we have $a\geq2\gamma-1$ and $b\geq\gamma$. We demonstrate that these lower bounds are tight, making use of them we provide a method to achieve the minimum size of ETSs belonging to irregular LDPC codes with girth 8 specially those whose column weight values are a subset of $\{2,3,4,5,6\}$. Moreover, we show for variable-regular LDPC codes with girth 10, $a\geq(\gamma-1)^2+1$. And for $\gamma=3,4$ we obtain $a\geq7$ and $a\geq12$, respectively. Finally, for variable-regular LDPC codes with girths $g=2(2k+1)$ and $g=2(2k+2)$ we obtain $a\geq(\gamma-2)^k+1$ and $a\geq2(\gamma-2)^k+1$, respectively., Comment: 17 pages, 5 figures

One of the serious issues in communication between people is hiding information from others, and the best way for this, is deceiving them. Since nowadays face images are mostly used in three dimensional format, in this paper we are going to steganography 3D face images, detecting which by curious people will be impossible. As in detecting face only its texture is important, we separate texture from shape matrices, for eliminating half of the extra information, steganography is done only for face texture, and for reconstructing 3D face, we can use any other shape. Moreover, we will indicate that, by using two textures, how two 3D faces can be combined. For a complete description of the process, first, 2D faces are used as an input for building 3D faces, and then 3D textures are hidden within other images., Comment: 6 pages, 10 figures, 16 equations, 5 sections

A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them.

LDPC lattices were the first family of lattices which have an efficient decoding algorithm in high dimensions over an AWGN channel. Considering Construction D' of lattices with one binary LDPC code as underlying code gives the well known Construction A LDPC lattices or 1-level LDPC lattices. Block-fading channel (BF) is a useful model for various wireless communication channels in both indoor and outdoor environments. Frequency-hopping schemes and orthogonal frequency division multiplexing (OFDM) can conveniently be modelled as block-fading channels. Applying lattices in this type of channel entails dividing a lattice point into multiple blocks such that fading is constant within a block but changes, independently, across blocks. The design of lattices for BF channels offers a challenging problem, which differs greatly from its counterparts like AWGN channels. Recently, the original binary Construction A for lattices, due to Forney, have been generalized to a lattice construction from totally real and complex multiplication fields. This generalized Construction A of lattices provides signal space diversity intrinsically, which is the main requirement for the signal sets designed for fading channels. In this paper we construct full diversity LDPC lattices for block-fading channels using Construction A over totally real number fields. We propose a new iterative decoding method for these family of lattices which has complexity that grows linearly in the dimension of the lattice. In order to implement our decoding algorithm, we propose the definition of a parity check matrix and Tanner graph for full diversity Construction A lattices. We also prove that the constructed LDPC lattices together with the proposed decoding method admit diversity order n-1 over an n-block-fading channel., Comment: 44 pages, 6 figures. Part of this work has been presented at ISIT 2016, Spain

Low density parity check (LDPC) lattices are obtained from Construction D' and a family of nested binary LDPC codes. We consider an special case of these lattices with one binary LDPC code as underlying code. This special case of LDPC lattices can be obtained by lifting binary LDPC codes using Construction A lattices. The LDPC lattices were the first family of lattices which have efficient decoding in high dimensions. We employ the encoding and decoding of the LDPC lattices in a cooperative transmission framework. We establish two efficient shaping methods based on hypercube shaping and Voronoi shaping, to obtain LDPC lattice codes. Then, we propose the implementation of block Markov encoding for one-way and two-way relay networks using LDPC lattice codes. This entails owning an efficient method for decomposing full-rate codebook into lower rate codebooks. We apply different decomposition schemes for one-way and two-way relay channels which are the altered versions of the decomposition methods of low density lattice codes (LDLCs). Due to the lower complexity of the decoding for LDPC lattices comparing to LDLCs, the complexity of our schemes are significantly lower than the ones proposed for LDLCs. The efficiency of the proposed schemes are presented using simulation results., Comment: 30 pages, 2 figures

LDPC lattices were the first family of lattices that equipped with iterative decoding algorithms under which they perform very well in high dimensions. In this paper, we introduce quasi cyclic low density parity check (QC-LDPC) lattices as a special case of LDPC lattices with one binary QC-LDPC code as their underlying code. These lattices are obtained from Construction A of lattices providing us to encode them efficiently using shift registers. To benefit from an encoder with linear complexity in dimension of the lattice, we obtain the generator matrix of these lattices in "quasi cyclic" form. We provide a low-complexity decoding algorithm of QC-LDPC lattices based on sum product algorithm. To design lattice codes, QC-LDPC lattices are combined with nested lattice shaping that uses the Voronoi region of a sublattice for code shaping. The shaping gain and shaping loss of our lattice codes with dimensions $40$, $50$ and $60$ using an optimal quantizer, are presented. Consequently, we establish a family of lattice codes that perform practically close to the sphere bound., Comment: Submitted for possible publication

A $\textit{sigma partitioning}$ of a graph $G$ is a partition of the vertices into sets $P_1, \ldots, P_k$ such that for every two adjacent vertices $u$ and $v$ there is an index $i$ such that $u$ and $v$ have different numbers of neighbors in $P_i$. The $\textit{ sigma number}$ of a graph $G$, denoted by $\sigma(G)$, is the minimum number $k$ such that $ G $ has a sigma partitioning $P_1, \ldots, P_k$. Also, a $\textit{ lucky labeling}$ of a graph $G$ is a function $ \ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means that $ x $ and $y$ are adjacent). The $\textit{ lucky number}$ of $ G $, denoted by $\eta(G)$, is the minimum number $k $ such that $ G $ has a lucky labeling $ \ell :V(G) \rightarrow \mathbb{N}_k$. It was conjectured in [Inform. Process. Lett., 112(4):109--112, 2012] that it is $ \mathbf{NP} $-complete to decide whether $ \eta(G)=2$ for a given 3-regular graph $G$. In this work, we prove this conjecture. Among other results, we give an upper bound of five for the sigma number of a uniformly random graph.

The regular number of a graph G denoted by reg(G) is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an open problem in A. Ganesan et al. (2012) [3] about the complexity of determining the regular number of graphs. We show that computation of the regular number for connected bipartite graphs is NP-hard. Furthermore, we show that, determining whether reg(G) = 2 for a given connected 3-colorable graph G is NP-complete. Also, we prove that a new variant of the Monotone Not-All-Equal 3-Sat problem is NP-complete., Comment: 9 pages, 2 figures, accepted for publication in "Graphs and Combinatorics"

The low-density parity-check (LDPC) lattices perform very well in high dimensions under generalized min-sum iterative decoding algorithm. In this work we focus on 1-level LDPC lattices. We show that these lattices are the same as lattices constructed based on Construction A and low-density lattice-code (LDLC) lattices. In spite of having slightly lower coding gain, 1-level regular LDPC lattices have remarkable performances. The lower complexity nature of the decoding algorithm for these type of lattices allows us to run it for higher dimensions easily. Our simulation results show that a 1-level LDPC lattice of size 10000 can work as close as 1.1 dB at normalized error probability (NEP) of $10^{-5}$.This can also be reported as 0.6 dB at symbol error rate (SER) of $10^{-5}$ with sum-product algorithm., Comment: 1 figure, submitted to IWCIT 2013

Low density generator matrix (LDGM) codes have an acceptable performance under iterative decoding algorithms. This idea is used to construct a class of lattices with relatively good performance and low encoding and decoding complexity. To construct such lattices, Construction D is applied to a set of generator vectors of a class of LDGM codes. Bounds on the minimum distance and the coding gain of the corresponding lattices and a corollary for the cross sections and projections of these lattices are provided. The progressive edge growth (PEG) algorithm is used to construct a class of binary codes to generate the corresponding lattice. Simulation results confirm the acceptable performance of these class of lattices.

In this paper a new class of lattices called turbo lattices is introduced and established. We use the lattice Construction D to produce turbo lattices. This method needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our basis codes. Therefore, a set of nested turbo codes based on nested interleavers (block interleavers) and nested convolutional codes is built. To this end, we employ both tail-biting and zero-tail convolutional codes. Using these codes, along with construction D, turbo lattices are created. Several properties of Construction D lattices and fundamental characteristics of turbo lattices including the minimum distance, coding gain and kissing number are investigated. Furthermore, a multi-stage turbo lattice decoding algorithm based on iterative turbo decoding algorithm is given. We show, by simulation, that turbo lattices attain good error performance within $\sim1.25 dB$ from capacity at block length of $n=1035$. Also an excellent performance of only $\sim.5 dB$ away from capacity at SER of $10^{-5}$ is achieved for size $n=10131$., Comment: Submitted to IEEE Trans. on Inform. Theory since Dec 2010

In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb{F}_q$. For the first time M\"{o}bius and R\'edei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of R\'edei functions an exact formula for the inverse function is derived. The cycle structure of R\'edei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers., Comment: Accepted to appear in AMC

The concept and existence of sphere-bound-achieving and capacity-achieving lattices has been explained on AWGN channels by Forney. LDPC lattices, introduced by Sadeghi, perform very well under iterative decoding algorithm. In this work, we focus on an ensemble of regular LDPC lattices. We produce and investigate an ensemble of LDPC lattices with known properties. It is shown that these lattices are sphere-bound-achieving and capacity-achieving. As byproducts we find the minimum distance, coding gain, kissing number and an upper bound for probability of error for this special ensemble of regular LDPC lattices., Comment: This paper has been withdrawn by the author due to an error