Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio in the case of communication over an additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information Theory, Aug 200

A STUDY OF LINEAR ERROR CORRECTING CODES

Since Shannon's ground-breaking work in 1948, there have been two main development streams of channel coding in approaching the limit of communication channels, namely classical coding theory which aims at designing codes with large minimum Hamming distance and probabilistic coding which places the emphasis on low complexity probabilistic decoding using long codes built from simple constituent codes. This work presents some further investigations in these two channel coding development streams. Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents and Mattson-Solomon polynomials, and are complementary to each other. The two methods generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and projective geometry codes. Their extension to non binary fields is shown to be straightforward. These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative decoding. It is also shown that for some of these codes, maximum likelihood performance may be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords of the dual code for each iteration. Following a property of the revolving-door combination generator, multi-threaded minimum Hamming distance computation algorithms are developed. Using these algorithms, the previously unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated. In addition, the highest minimum Hamming distance attainable by all binary cyclic codes of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes which have higher minimum Hamming distance than the previously considered best known linear code have been found. It is shown that by exploiting the structure of circulant matrices, the number of codewords required, to compute the minimum Hamming distance and the number of codewords of a given Hamming weight of binary double-circulant codes based on primes, may be reduced. A means of independently verifying the exhaustively computed number of codewords of a given Hamming weight of these double-circulant codes is developed and in coiyunction with this, it is proved that some published results are incorrect and the correct weight spectra are presented. Moreover, it is shown that it is possible to estimate the minimum Hamming distance of this family of prime-based double-circulant codes. It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection mechanism that offers much better throughput and performance than the conventional ORG scheme is described. Using the same method it is shown that the performance of conventional CRC scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy communications system and it is shown that sequences of good error correction codes, suitable for use in incremental redundancy communications systems may be obtained using the Constructions X and XX. Examples are given and their performances presented in comparison to conventional CRC schemes

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

Exchange of Correlated Binary Sources in Two-Way Relay Networks Using LDPC Codes

We consider the problem of exchanging messages in two-way relay (TWR) systems when the sources are correlated binary sequences. In a TWR system, two users communicate simultaneously in both directions to exchange their messages with the help of a relay. Harnessing the fact that the users have access to their own non-compressed messages as side information, each user can compress its message according to the Slepian-Wolf coding strategy by using low-density parity-check codes, particularly, the syndrome approach. Through numerical examples, we show that the proposed scheme offers significant improvements in compression rates compared to the existing solutions in the literature. © 2012 IEEE

Improving Group Integrity of Tags in RFID Systems

Checking the integrity of groups containing radio frequency identification (RFID) tagged objects or recovering the tag identifiers of missing objects is important in many activities. Several autonomous checking methods have been proposed for increasing the capability of recovering missing tag identifiers without external systems. This has been achieved by treating a group of tag identifiers (IDs) as packet symbols encoded and decoded in a way similar to that in binary erasure channels (BECs). Redundant data are required to be written into the limited memory space of RFID tags in order to enable the decoding process. In this thesis, the group integrity of passive tags in RFID systems is specifically targeted, with novel mechanisms being proposed to improve upon the current state of the art. Due to the sparseness property of low density parity check (LDPC) codes and the mitigation of the progressive edge-growth (PEG) method for short cycles, the research is begun with the use of the PEG method in RFID systems to construct the parity check matrix of LDPC codes in order to increase the recovery capabilities with reduced memory consumption. It is shown that the PEG-based method achieves significant recovery enhancements compared to other methods with the same or less memory overheads. The decoding complexity of the PEG-based LDPC codes is optimised using an improved hybrid iterative/Gaussian decoding algorithm which includes an early stopping criterion. The relative complexities of the improved algorithm are extensively analysed and evaluated, both in terms of decoding time and the number of operations required. It is demonstrated that the improved algorithm considerably reduces the operational complexity and thus the time of the full Gaussian decoding algorithm for small to medium amounts of missing tags. The joint use of the two decoding components is also adapted in order to avoid the iterative decoding when the missing amount is larger than a threshold. The optimum value of the threshold value is investigated through empirical analysis. It is shown that the adaptive algorithm is very efficient in decreasing the average decoding time of the improved algorithm for large amounts of missing tags where the iterative decoding fails to recover any missing tag. The recovery performances of various short-length irregular PEG-based LDPC codes constructed with different variable degree sequences are analysed and evaluated. It is demonstrated that the irregular codes exhibit significant recovery enhancements compared to the regular ones in the region where the iterative decoding is successful. However, their performances are degraded in the region where the iterative decoding can recover some missing tags. Finally, a novel protocol called the Redundant Information Collection (RIC) protocol is designed to filter and collect redundant tag information. It is based on a Bloom filter (BF) that efficiently filters the redundant tag information at the tag’s side, thereby considerably decreasing the communication cost and consequently, the collection time. It is shown that the novel protocol outperforms existing possible solutions by saving from 37% to 84% of the collection time, which is nearly four times the lower bound. This characteristic makes the RIC protocol a promising candidate for collecting redundant tag information in the group integrity of tags in RFID systems and other similar ones

Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 221-238).Quantum computers have been shown to efficiently solve a class of problems for which no efficient solution is otherwise known. Physical systems can implement quantum computation, but devising realistic schemes is an extremely challenging problem largely due to the effect of noise. A quantum computer that is capable of correctly solving problems more rapidly than modern digital computers requires some use of so-called fault-tolerant components. Code-based fault-tolerance using quantum error-correcting codes is one of the most promising and versatile of the known routes for fault-tolerant quantum computation. This dissertation presents three main, new results about code-based fault-tolerant quantum computer architectures. The first result is a large new family of quantum codes that go beyond stabilizer codes, the most well-studied family of quantum codes. Our new family of codeword stabilized codes contains all known codes with optimal parameters. Furthermore, we show how to systematically find, construct, and understand such codes as a pair of codes: an additive quantum code and a classical (nonlinear) code. Second, we resolve an open question about universality of so-called transversal gates acting on stabilizer codes. Such gates are universal for classical fault-tolerant computation, but they were conjectured to be insufficient for universal fault-tolerant quantum computation. We show that transversal gates have a restricted form and prove that some important families of them cannot be quantum universal. This is strong evidence that so-called quantum software is necessary to achieve universality, and, therefore, fault-tolerant quantum computer architecture is fundamentally different from classical computer architecture. Finally, we partition the fault-tolerant design problem into levels of a hierarchy of concatenated codes and present methods, compatible with rigorous threshold theorems, for numerically evaluating these codes.(cont.) The methods are applied to measure inner error-correcting code performance, as a first step toward elucidation of an effective fault-tolerant quantum computer architecture that uses no more than a physical, inner, and outer level of coding. Of the inner codes, the Golay code gives the highest pseudothreshold of 2 x 10-3. A comparison of logical error rate and overhead shows that the Bacon-Shor codes are competitive with Knill's C₄/C₆ scheme at a base error rate of 10⁻⁴.by Andrew W. Cross.Ph.D