Code Graph Research Articles

Sparse Graph Codes for Channels with Synchronous Errors

Sparse Graph Codes for Channels with Synchronous Errors

Additive perfect codes in Doob graphs

The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m \times (Z_2^2)^{n'} \times Z_4^{n''}$, where $n'+n''=n$. A set of vertices of $D(m,n)$ is called an additive code if it forms a subgroup of this group. We construct a $3$-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive $1$-perfect codes in $D(m,n'+n'')$ are sufficient. Additionally, two quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and $D(2667,0+127)$.

Fast Beam Alignment for Millimeter Wave Communications: A Sparse Encoding and Phaseless Decoding Approach

In this paper, we studied the problem of beam alignment for millimeter wave (mmWave) communications, in which we assume a hybrid analog and digital beamforming structure is employed at the transmitter (i.e. base station), and an omni-directional antenna or an antenna array is used at the receiver (i.e. user). By exploiting the sparse scattering nature of mmWave channels, the beam alignment problem is formulated as a sparse encoding and phaseless decoding problem. More specifically, the problem of interest involves finding a sparse sensing matrix and an efficient recovery algorithm to recover the support and magnitude of the sparse signal from compressive phaseless measurements. A sparse bipartite graph coding (SBG-Coding) algorithm is developed for sparse encoding and phaseless decoding. Our theoretical analysis shows that, in the noiseless case, our proposed algorithm can perfectly recover the support and magnitude of the sparse signal with probability exceeding a pre-specified value from $\mathcal{O}(K^2)$ measurements, where $K$ is the number of nonzero entries of the sparse signal. The proposed algorithm has a simple decoding procedure which is computationally efficient and noise-robust. Simulation results show that our proposed method renders a reliable beam alignment in the low and moderate signal-to-noise ratio (SNR) regimes and presents a clear performance advantage over existing methods.

On non-commutative operator graphs generated by covariant resolutions of identity

We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.

Polar Codes for Informed Receivers

The problem of sending independent messages to a receiver which already has been informed a subset of messages as side information is studied. Which messages are available at the receiver is assumed to be oblivious to the transmitter. A simple yet powerful construction of codes for informed receivers based on polar codes is proposed to efficiently and fairly convert side information into reduction in probability of errors for any message side information. Compared with existing designs based on codes on graph or algebraic codes, the proposed construction is both conceptually and practically simple. Moreover, the codes inherit many good properties of polar codes, including explicit construction and low-complexity encoding and decoding scaling such as $\mathcal {O}(N\log N)$ where $N$ is the blocklength. Extensions to high-order modulation and block fading channels are also proposed. Simulation results indicate that for every side information configuration, the proposed codes can convert side information into reduction in signal-to-noise ratio of roughly 6 dB/bit at bit error rate of 10−5 under additive white Gaussian noise channel or block fading channel.

Optimal bounds on codes for location in circulant graphs

Identifying and locating-dominating codes have been studied widely in circulant graphs of type $C_{n}(1,2,3,\dots , r)$ over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs $C_{n}(1,d)$ for $d = 3$ and proposed as an open question the case of $d > 3$ . In this paper we study identifying, locating-dominating and self-identifying codes in the graphs $C_{n}(1,d)$ , $C_{n}(1,d-1,d)$ and $C_{n}(1,d-1,d,d + 1)$ . We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in $C_{n}(1,3)$ and $C_{n}(1,4)$ .

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.

PEG-Like Design of Binary QC-LDPC Codes Based on Detecting and Avoiding Generating Small Cycles

In this paper, we propose a new multi-edge metric-constrained quasi-cyclic progressive edge-growth algorithm (MM-QC-PEGA), which is suitable for constructing both single- and multi-weighted (binary) QC low-density parity-check (LDPC) codes with arbitrary lengths, rates, circulant sizes, and variable node (VN)-degree distributions. The MM-QC-PEGA is able to detect all cyclic-edge-set-minimum-virtual cycles (CMVCs), as it accurately computes the metric value of each CMVC with an a posteriori test. In addition, we propose a new greatest-common-divisor (GCD)-approximation for time efficiently approximating the metric value of a CMVC without a posteriori tests, and propose a new GCD-approximated MM-QC-PEGA (G-MM-QC-PEGA) by using the GCD-approximation to replace the a posteriori test involved in the MM-QC-PEGA. As a result, the G-MM-QC-PEGA is faster than the MM-QC-PEGA, and the CMVCs undetectable to the G-MM-QC-PEGA under multi- and single-weighted QC-LDPC code graphs have minimum lengths of eight and ten, respectively. Moreover, we propose a new masking technique that is efficient in masking the parity-check matrix consisting of an array of arbitrary circulants of the same size. Compared with the several existing works, our proposed algorithms could perform better or comparably in terms of avoiding generating small cycles and error performances. Our proposed algorithms somewhat perfect the works of QC-LDPC code construction.

Analysis of quantum error correction with symmetric hypergraph states

Graph states have been used to construct quantum error correction codes for independent errors. Hypergraph states generalize graph states, and symmetric hypergraph states have been shown to allow for the correction of correlated errors. In this paper, it is shown that symmetric hypergraph states are not useful for the correction of independent errors, at least for up to 30 qubits. Furthermore, error correction for error models with protected qubits is explored. A class of known graph codes for this scenario is generalized to hypergraph codes.

Perfect Codes in Cayley Graphs

Given a graph $\Gamma$, a subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$, and a subset $C$ of $V(\Gamma)$ is called a total perfect code in $\Gamma$ if every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.

A New Multi-Edge Metric-Constrained PEG Algorithm for Designing Binary LDPC Code With Improved Cycle-Structure

To obtain a better cycle-structure is still a challenge for the low-density parity-check (LDPC) code design. This paper formulates two metrics firstly so that the progressive edge-growth (PEG) algorithm and the approximate cycle extrinsic message degree (ACE) constrained PEG algorithm are unified into one integrated algorithm, called the metric-constrained PEG algorithm (M-PEGA). Then, as an improvement for the M-PEGA, the multi-edge metric-constrained PEG algorithm (MM-PEGA) is proposed based on two new concepts, the multi-edge local girth and the edge-trials. The MM-PEGA with the edge-trials, say a positive integer $r$, is called the $r$-edge M-PEGA, which constructs each edge of the non-quasi-cyclic (non-QC) LDPC code graph through selecting a check node whose $r$-edge local girth is optimal. In addition, to design the QC-LDPC codes with any predefined valid design parameters, as well as to detect and even to avoid generating the undetectable cycles in the QC-LDPC codes designed by the QC-PEG algorithm, the multi-edge metric constrained QC-PEG algorithm (MM-QC-PEGA) is proposed lastly. It is verified by the simulation results that increasing the edge-trials of the MM-PEGA/MM-QC-PEGA is expected to have a positive effect on the cycle-structures and the error performances of the LDPC codes designed by the MM-PEGA/MM-QC-PEGA.

Выявление вредоносного программного обеспечения на основе классификации графов потоков исходных кодов

Выявление вредоносного программного обеспечения на основе классификации графов потоков исходных кодов

Optimized Construction of Protograph G-LDPC Codes by Modified EXIT Chart and MACE for New-Generation Wireless Communications

In this paper, a new systematic construction of optimized protograph generalized low-density parity-check codes with good decoding threshold and low error floor is proposed. First, a typical code graph is generated by combining an accumulate-repeat-accumulate seed protograph with a Tanner graph extension of a simple linear block node. Subsequently, it is analyzed and optimized theoretically, especially with the puncture mechanism, by a modified extrinsic information transfer chart and an asymptotic weight distribution. The generated graph is then extended to a base matrix by a copy-and-permute procedure, accompanied with the matrix split, which is optimized by a progressive edge growth for good randomness and girth property. Finally, the proposed code matrix is created by replacing “1” in the above base matrix with square circulant sub-matrices, the offsets of which are searched by a quasi-cyclic (QC)-oriented modified approximate cycle extrinsic message degree algorithm to improve the cycle relationship, especially for the compound cycles. Simulation results show that the codes exhibit excellent performance in both error floor and waterfall region on an additive white Gaussian noise channel. Moreover, they are also characterized by faster encoding due to the QC structure as well as lower decoding complexity and less latency, which make them a natural fit for new-generation power constrained wireless communications.

Perfect codes in power graphs of finite groups

Abstract The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

Video distortion reduction with instantly decodable network coding and Device-to-Device communications

In this paper, we propose a scheme of wireless multimedia network coded broadcast retransmission, based on Device-to-Device (D2D) communications, in which there are network coding conflicts and transmission conflicts between multiple terminal devices. For this reason, a novel instantly decodable network coding (IDNC) graph is constructed, where all the feasible coding and non-transmission collision cases are shown in the graph. Based on the constructed IDNC graph, we design the optimal broadcast strategy for batch transmission and sliding window transmission, and propose a heuristic maximum weight network coding selection algorithm (MWSA-NC) to reduce the complexity of the optimal search algorithm. The simulation results show that compared to the traditional scheme, the proposed algorithm can greatly improve the throughput of the system, reduce decoding latency and reduce video stream distortion.

A linear-time algorithm for the identifying code problem on block graphs

The identifying code problem is a special search problem, challenging both from a theoretical and from a computational point of view, even for several graphs where other usually hard problems are easy to solve, like bipartite graphs or chordal graphs. Hence, a typical line of attack for this problem is to determine minimum identifying codes of special graphs. In this work we study the problem of determining the cardinality of a minimum identifying code in block graphs (that are diamond-free chordal graphs). We present a linear-time algorithm for this problem, as a generalization of a linear-time algorithm proposed by Auger in 2010 for the case of trees. Thereby, we provide a subclass of chordal graphs for which the identifying code problem can be solved in linear time.

Polyhedra associated with identifying codes in graphs

The identifying code problem is a newly emerging search problem, challenging both from a theoretical and a computational point of view, even for special graphs like bipartite graphs. Hence, a typical line of attack for this problem is to determine minimum identifying codes of special graphs or to provide bounds for their size.In this work we study the associated polyhedra and present some general results on their combinatorial structure. We demonstrate how the polyhedral approach can be applied to find minimum identifying codes for special graphs, and discuss further lines of research in order to obtain strong lower bounds stemming from linear relaxations of the identifying code polyhedron, enhanced by suitable cutting planes to be used in a B&C framework.

On Minimum Identifying Codes in Some Cartesian Product Graphs

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code, or ID code, in a graph G is called the ID code number of G and is denoted $$\gamma ^\mathrm{ID}(G)$$ . In this paper, we give upper and lower bounds for the ID code number of the prism of a graph, or $$G\Box K_2$$ . In particular, we show that $$\gamma ^\mathrm{ID}(G \Box K_2) \ge \gamma ^\mathrm{ID}(G)$$ and we show that this bound is sharp. We also give upper and lower bounds for the ID code number of grid graphs and a general upper bound for $$\gamma ^\mathrm{ID}(G\Box K_2)$$ .

MDS codes in Doob graphs

The Doob graph $D(m,n)$, where $m>0$, is the direct product of $m$ copies of The Shrikhande graph and $n$ copies of the complete graph $K_4$ on $4$ vertices. The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. In this paper we consider MDS codes in Doob graphs with code distance $d \ge 3$. We prove that if $2m+n>6$ and $2<d<2m+n$, then there are no MDS codes with code distance $d$. We characterize all MDS codes with code distance $d \ge 3$ in Doob graphs $D(m,n)$ when $2m+n \le 6$. We characterize all MDS codes in $D(m,n)$ with code distance $d=2m+n$ for all values of $m$ and $n$.

Edge-Based Dynamic Scheduling for Belief-Propagation Decoding of LDPC and RS Codes

This paper presents two low-complexity edge-based scheduling schemes, referred to as the e-Flooding and e-Shuffled schedules, for the belief-propagation (BP) decoding of low-density parity-check and Reed–Solomon codes. The proposed schedules selectively update the edges of the code graph based on the run-time reliability of variable and check nodes. Specifically, new message update is propagated exclusively along the unreliable edges of the code graph. This reduces the decoding complexity of BP algorithm as only a partial set of message updates is computed per decoding iteration. Besides, restricting the flow of message updates may also precludes the occurrence of some short graph cycles, which helps to preserve the BP message independence at certain variable and check nodes. Using numerical simulations, it is shown that the proposed edge-based schedules reduce the BP decoding complexity by more than 90% compared with the prior-art BP schedules, while simultaneously improving the error-rate performance, at medium-to-high signal-to-noise ratio over additive white Gaussian noise channel.