Code Graph Research Articles

Identifying and Locating–Dominating Codes in (Random) Geometric Networks

We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.

Transactions papers evaluation and design of irregular LDPC codes using ACE spectrum

The construction of finite-length irregular LDPC codes with low error floors is currently an attractive research problem. In particular, for the binary erasure channel (BEC), the problem is to find the elements of selected irregular LDPC code ensembles with the size of their minimum stopping set being maximized. Due to the lack of analytical solutions to this problem, a simple but powerful heuristic design algorithm, the approximate cycle extrinsic message degree (ACE) constrained design algorithm, has recently been proposed. Building upon the ACE metric associated with a cycle in a code graph, we introduce the ACE spectrum of LDPC codes as a useful tool for evaluation of codes from selected irregular LDPC code ensembles. Using the ACE spectrum, we generalize the ACE constrained design algorithm, making it more flexible and efficient. We justify the ACE spectrum approach through examples and simulation results.

Identifying Codes in Trees and Planar Graphs

Abstract We deal with a few issues related to the problem of finding the minimum size of an identifying code in a graph. First, we provide a linear algorithm which computes an identifying code with minimal size in a given tree. Second, we extend known NP-hardness results by showing that this problem remains NP-hard in the class of planar graphs with arbitrary high girth and maximal degree at most 4. We give similar results for the problem of finding the minimum size of an ( r , ⩽ l ) -identifying code, for all r ⩾ 1 and l ∈ { 1 ; 2 } .

Low-Density Graph Codes That Are Optimal for Binning and Coding With Side Information

In this paper, we describe and analyze the source and channel coding properties of a class of sparse graphical codes based on compounding a low-density generator matrix (LDGM) code with a low-density parity-check (LDPC) code. Our first pair of theorems establishes that there exist codes from this ensemble, with all degrees remaining bounded independently of block length, that are simultaneously optimal for both channel coding and source coding with binary data when encoding and decoding are performed optimally. More precisely, in the context of lossy compression, we prove that finite-degree constructions can achieve any pair (R, D) on the rate-distortion curve of the binary symmetric source. In the context of channel coding, we prove that the same finite-degree codes can achieve any pair (C, p) on the capacity-noise curve of the binary symmetric channel (BSC). Next, we show that our compound construction has a nested structure that can be exploited to achieve the Wyner-Ziv bound for source coding with side information (SCSI), as well as the Gelfand-Pinsker bound for channel coding with side information (CCSI). Although the results described here are based on optimal encoding and decoding, the proposed graphical codes have sparse structure and high girth that renders them well suited to message passing and other efficient decoding procedures.

Graph-based classification of self-dual additive codes over finite fields

Quantum stabilizer states over Fm can be represented as self-dual additive codes over Fm2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operationcorrespond to equivalence classes of codes. We have previously used this fact to classify self-dual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDSconjecture holds, we are able to classify all self-dual additive MDS codes over F9 by using an extension technique. We prove that the minimum distanceof a self-dual additive code is related to the minimum vertex degree in theassociated graph orbit. Circulant graph codes are introduced, and a computersearch reveals that this set contains many strong codes. We show that some ofthese codes have highly regular graph representations.

Quantum-error-correcting codes using qudit graph states

Graph states are generalized from qubits to collections of $n$ qudits of arbitrary dimension $D$, and simple graphical methods are used to construct both additive and nonadditive, as well as degenerate and nondegenerate, quantum-error-correcting codes. Codes of distance 2 saturating the quantum Singleton bound for arbitrarily large $n$ and $D$ are constructed using simple graphs, except when $n$ is odd and $D$ is even. Computer searches have produced a number of codes with distances 3 and 4, some previously known and some new. The concept of a stabilizer is extended to general $D$, and shown to provide a dual representation of an additive graph code.

Counter braids

Fine-grained network measurement requires routers and switches to update large arrays of counters at very high link speed (e.g. 40 Gbps). A naive algorithm needs an infeasible amount of SRAM to store both the counters and a flow-to-counter association rule, so that arriving packets can update corresponding counters at link speed. This has made accurate per-flow measurement complex and expensive, and motivated approximate methods that detect and measure only the large flows. This paper revisits the problem of accurate per-flow measurement. We present a counter architecture, called Counter Braids, inspired by sparse random graph codes. In a nutshell, Counter Braids "compresses while counting". It solves the central problems (counter space and flow-to-counter association) of per-flow measurement by "braiding" a hierarchy of counters with random graphs. Braiding results in drastic space reduction by sharing counters among flows; and using random graphs generated on-the-fly with hash functions avoids the storage of flow-to-counter association. The Counter Braids architecture is optimal (albeit with a complex decoder) as it achieves the maximum compression rate asymptotically. For implementation, we present a low-complexity message passing decoding algorithm, which can recover flow sizes with essentially zero error. Evaluation on Internet traces demonstrates that almost all flow sizes are recovered exactly with only a few bits of counter space per flow.

Woven Graph Codes: Asymptotic Performances and Examples

Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensemble of such codes based on $s$-partite, $s$-uniform hypergraphs, where $s$ depends only on the code rate, there exist codes satisfying the Varshamov-Gilbert (VG) and the Costello lower bound on the minimum distance and the free distance, respectively. A connection between regular bipartite graphs and tailbiting codes is shown. Some examples of woven graph codes are presented. Among them an example of a rate $R_{\rm wg}=1/3$ woven graph code with $d_{\rm free}=32$ based on Heawood's bipartite graph and containing $n=7$ constituent rate $R^{c}=2/3$ convolutional codes with overall constraint lengths $\nu^{c}=5$ is given. An encoding procedure for woven graph codes with complexity proportional to the number of constituent codes and their overall constraint length $\nu^{c}$ is presented.

Estimating random variables from random sparse observations

AbstractLet X1, … , Xn, be a collection of iid discrete random variables, and Y1, … , Ym, a set of noisy observations of such variables. Assume each observation Ya, to be a random function of a random subset of the Xi,s, and consider the conditional distribution of Xi, given the observations, namely µi,(xi,) ≡ ${\cal P}${Xi, = xi,|Y} (a posteriori probability).We establish a general decoupling principle among the Xi,s, as well as a relation between the distribution of µi, and the fixed points of the associated density evolution operator. These results hold asymptotically in the large system limit, provided the average number of variables an observation depends on is bounded. We discuss the relevance of our result to a number of applications, ranging from sparse graph codes and multi‐user detection, to group testing. Copyright © 2008 John Wiley & Sons, Ltd.

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the eva- luation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC� ) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date.

Generalized LDPC codes and generalized stopping sets

A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.

Discriminating codes in bipartite graphs: bounds, extremal cardinalities, complexity

Consider an undirected bipartite graph $G=(V=I\cup A,E)$, with no edge inside $I$ nor $A$. For any vertex $v\in V$, let $N(v)$ be the set of neighbours of $v$. A code $C \subseteq A$ is said to be discriminating if all the sets $N(i) \cap C$, $i \in I$, are nonempty and distinct. We study some properties of discriminating codes. In particular, we give bounds on the minimum size of these codes, investigate graphs where minimal discriminating codes have size close to the upper bound, or give the exact minimum size in particular graphs; we also give an NP-completeness result.

Generalized ACE Constrained Progressive Edge-Growth LDPC Code Design

In this letter, we propose a generalization of the progressive edge-growth (PEG) algorithm with the aim of designing LDPC code graphs with substantially improved approximated cycle extrinsic message degree (ACE) properties. The proposed realization of generalized PEG algorithm outperforms original PEG algorithm and its subsequent modification proposed by Xiao and Banihashemi.

The Development of Turbo and LDPC Codes for Deep-Space Applications

The development of error-correcting codes has been closely coupled with deep-space exploration since the early days of both. Since the discovery of turbo codes in 1993, the research community has invested a great deal of work on modern iteratively decoded codes, and naturally NASA's Jet Propulsion Laboratory (JPL) has been very much involved. This paper describes the research, design, implementation, and standardization work that has taken place at JPL for both turbo and low-density parity-check (LDPC) codes. Turbo code development proceeded from theoretical analyses of polynomial selection, weight distributions imposed by interleaver designs, decoder error floors, and iterative decoding thresholds. A family of turbo codes was standardized and implemented and is currently in use by several spacecraft. JPL's LDPC codes are built from protographs and circulants, selected by analyses of decoding thresholds and methods to avoid loops in the code graph. LDPC encoders and decoders have been implemented in hardware for planned spacecraft, and standardization is under way.

On optimal codes with w-identifiable parent property

Let $$\mathcal{C}$$ be a q-ary code of length n and $$X \subseteq \mathcal{C}$$ , then d is called a descendant of X if d i ? {x i : x ?X} for all 1 ? i ? n. $$\mathcal{C}$$ is said to be a w-identifiable parent property code (w-IPP code for short) if whenever d is a descendant of w (or fewer) codewords, one can always identify at least one of the parent codewords in $$\mathcal{C}$$ . In this paper, we give constructions for w-IPP codes of length w + 1. Furthermore, we show that F w (w + 1,q), the maximum cardinality of a w-IPP q-ary code of length w + 1, satisfies $$|\fancyscript{G}_{h(q)}| \leq F_{w}(w+1,q) \leq |\fancyscript{G}_{h(q)}|+6$$ , where $$\fancyscript{G}_{h(q)}$$ is a well-defined code graph. Finally, we give an efficient (O(q w+1)) algorithm to find the values of F w (w + 1,q).

Augmented Belief Propagation Decoding of Low-Density Parity Check Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We propose an augmented belief propagation (BP) decoder for low-density parity check (LDPC) codes which can be utilized on memoryless or intersymbol interference channels. The proposed method is a heuristic algorithm that eliminates a large number of pseudocodewords that can cause nonconvergence in the BP decoder. The augmented decoder is a multistage iterative decoder, where, at each stage, the original channel messages on select symbol nodes are replaced by saturated messages. The key element of the proposed method is the symbol selection process, which is based on the appropriately defined subgraphs of the code graph and/or the reliability of the information received from the channel. We demonstrate by examples that this decoder can be implemented to achieve substantial gains (compared to the standard locally-operating BP decoder) for short LDPC codes decoded on both memoryless and intersymbol interference Gaussian channels. Using the Margulis code example, we also show that the augmented decoder reduces the error floors. Finally, we discuss types of BP decoding errors and relate them to the augmented BP decoder. </para>

Generalized Low-Density Parity-Check Codes Based on Hadamard Constraints

In this paper, we consider the design and analysis of generalized low-density parity-check (GLDPC) codes in AWGN channels. The GLDPC codes are specified by a bipartite Tanner graph, as with standard LDPC codes, but with the single parity-check constraints replaced by general coding constraints. In particular, we consider imposing Hadamard code constraints at the check nodes for a low-rate approach, termed LDPC-Hadamard codes. We introduce a low-complexity message-passing based iterative soft-input soft-output (SISO) decoding algorithm, which employs the a posteriori probability (APP) fast Hadamard transform (FHT) for decoding the Hadamard check codes at each decoding iteration. The achievable capacity with the GLDPC codes is then discussed. A modified LDPC-Hadamard code graph is also proposed. We then optimize the LDPC-Hadamard code ensemble using a low-complexity optimization method based on approximating the density evolution by a one-dimensional dynamic system represented by an extrinsic mutual information transfer (EXIT) chart. Simulation results show that the optimized LDPC-Hadamard codes offer better performance in the low-rate region than low-rate turbo-Hadamard codes, but also enjoy a fast convergence rate. A rate-0.003 LDPC-Hadamard code with large block length can achieve a bit-error-rate (BER) performance of 10-5 at -1.44 dB, which is only 0.15 dB away from the ultimate Shannon limit (-1.592 dB) and 0.24 dB better than the best performing low-rate turbo-Hadamard codes

Codes from the line graphs of complete multipartite graphs and PD-sets

The binary codes of the line graphs L m ( n ) of the complete multipartite graphs K n 1 , … , n m ( n i = n for 1 ⩽ i ⩽ m ) n ⩾ 2 , m ⩾ 3 are examined, and PD-sets and s-PD-sets are found.

Construction of Regular and Irregular LDPC Codes: Geometry Decomposition and Masking

Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs

On a new class of identifying codes in graphs

Assume that G = ( V , E ) is an undirected graph, and C ⊆ V . For every v ∈ V , we denote I r ( v ) = { u ∈ C : d ( u , v ) ⩽ r } , where d ( u , v ) denotes the number of edges on any shortest path from u to v. For every F ⊆ V , we denote I r ( F ) = ⋃ v ∈ F I r ( v ) . We study codes C with the property that if I r ( F ) = I r ( F ′ ) and F ≠ F ′ , then both F and F ′ have size at least l + 1 . Such codes can be used in the maintenance of multiprocessor architectures. We consider the cases when G is the infinite square or king grid, infinite triangular lattice or hexagonal mesh, or a binary hypercube.