Topics in Discrete Mathematics: Error-Correcting Codes

2008

Cooperative diversity represents a new class of wireless communication techniques in which network nodes help each other in relaying information to realize spatial diversity advantages. This new transmission paradigm promises significant performance gains in terms of link reliability, spectral efficiency, system capacity, and transmission range. Cooperative diversity has spurred tremendous excitement within the academia and industry circles since its introduction and has been extensively studied over the last few years.

Lecture Notes in Computer Science, 1999

VI Preface AAECC-13 received 86 submissions; 42 were selected for publication in these proceedings while 33 additional works will contribute to the symposium as oral presentations.

1999

VI Preface AAECC-13 received 86 submissions; 42 were selected for publication in these proceedings while 33 additional works will contribute to the symposium as oral presentations.

2007

In this paper, codes which reduce the peak-to-average power ratio (PAPR) in multi-code code division multiple access (MC-CDMA) communication systems are studied. It is known that using bent functions to define binary codewords gives constant amplitude signals. Based on the concept of quarter bent functions, a new inequality relating the minimum order of terms of a bent function and the maximum Walsh spectral magnitude is proved, and it facilitates the generalization of some known results.

2012

Computing Research Repository, 2011

This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse - a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n\times n-sensing matrices contain, on average, \Omega(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the above results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Finally, we provide a non-exhaustive procedure to search for the unknown low-rank matrix.

IEEE Transactions on Information Theory, 2000

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

IEEE Transactions on Communications, 2000

The Lee metric measures the circular distance between two elements in a cyclic group and is particularly appropriate as a measure of distance for data transmission under phase-shift keying modulation over a white noise channel. In this paper, using newly derived properties on Newton's identities, we initially investigate the Lee distance properties of a class of BCH codes and we show that (for an appropriate range of parameters) their minimum Lee distance is at least twice their designed Hamming distance. We then make use of properties of these codes to devise an efficient algebraic decoding algorithm that successfully decodes within the above lower bound of the Lee error-correction capability. Finally, we propose an attractive design for the corresponding VLSI architecture that is only mildly more complex than popular decoder architectures under the Hamming metric; since the proposed architecture can be re-used for decoding under the Hamming metric without extra hardware, one can use the proposed architecture to decode under both distance metrics (Lee and Hamming).