Quantum memory is essential for the development of many devices in quantum information processing, including a synchronization tool that can match the various processes in a quantum computer, a quantum gate that maintains the identity of any state, and a mechanism for converting predetermined photons into on-demand photons. Quantum memory can be used in many aspects, such as quantum computing and quantum communication. Continuous research and experiments have enabled quantum memory to realize the storage of qubits. On the other hand, interleaving techniques are used to deal with burst of errors. Many works from literature provide effective interleaving techniques for combating burst of errors by using classical error-correcting codes. However, to the best of our knowledge, little is known regarding interleaving techniques for combating cluster of errors in topological quantum error-correcting codes. Thenceforward, it is proposed in this work a new interleaving technique by using good toric quantum codes, namely Bombin and Martin-Delgado toric codes and another class of toric codes from literature which is described in this work. As a consequence, by applying this new interleaving method to such toric quantum codes, new quantum burst-error correcting codes are obtained which improve the coding gain. In addition to the proposed quantum interleaving technique improves such parameter, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory.

In order to understand how biological information within DNA and RNA biosignals is preserved, several works have proposed that these biosequences can be identified as codewords of BCH error correcting codes over GF(4) (F4). The current mathematical tool, used to perform such identification, has some limitations: one of them being the need of knowing all binary primitive polynomials and the other one is essentially due to the constraint imposed by the cyclic code construction leading to a restriction regarding the nucleotide sequence lengths. In this work, a novel algorithm for identifying odd-length nucleotide sequences as codewords of BCH codes over GF(4) is presented. As a result more than 270 cDNA sequences of nine different lengths, which could not be considered by the previous algorithm, are identified as codewords of 35 BCH codes.

The problem of searching for perfect codes has attracted great attention since the paper by Golomb and Welch, in which the existence of these codes over Lee metric spaces was considered. Since perfect codes are not very common, the problem of searching for quasi-perfect codes is also of great interest. In this aspect, also quasi-perfect Lee codes have been considered for 2-D and 3-D Lee metric spaces. In this paper, constructive methods for obtaining quasi-perfect codes over metric spaces modeled by means of Gaussian and Eisenstein-Jacobi integers are given. The obtained codes form ideals of the integer ring thus preserving the property of being geometrically uniform codes. Moreover, they are able to correct more error patterns than the perfect codes which may properly be used in asymmetric channels. Therefore, the results in this paper complement the constructions of perfect codes previously done for the same integer rings. Finally, decoding algorithms for the quasi-perfect codes obtained in this paper are provided and the relationship of the codes and the Lee metric ones is investigated.

In this paper we propose a binary arithmetic coding technique for encoding the output of chaotic sources. The model we use for such sources is the free motion of particles on a surface of constant negative curvature. The source symbol probabilities are time-varying and follow a well defined rule as the one associated with a fractal known as Maskit boundary. We show that the average codeword lengths resulting from the proposed technique is less than the length of the original sequences.

In this paper we present classes of topological quantum codes (TQC) derived from the embedding of complete bipartite graphs on the corresponding compact surfaces. Every code achieves minimum distance 3 and its encoding rate is such that k/n → 1 as n → ∞.

In this paper we present a generalization of the perfect codes derived from the quotient rings of Gaussian integers. We call this class of codes quasi-perfect, which in addition to preserving the property of being geometrically uniform codes they are able to correct more error patterns than the perfect codes.

One of the puzzling problems in mathematical biology is to show the existence of any form of error-correcting code in the DNA structure. Here we propose a model for the biological coding system similar to that of a digital communication system. This model consists of an encoder (a mapper and a BCH code over ℤ4) and a modulator (genetic code). Here we show that DNA sequences including proteins and targeting sequences from different species with 63, 255, and 1023 nucleotides long were identified as codewords of ℤ4-linear codes.

In this paper we construct a class of topological quantum error-correcting codes on compact surfaces with genus g ≥ 2. This generalizes the construction process of toric codes. As examples, we tabulate some good topological quantum codes on surfaces with genus 2 and 3.

We construct a concatenated [(4, 1, 3)] quantum convolutional code from a (2, 1, 2) classical convolutional code. This quantum convolutional code can correct one general error.

Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney . Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered.

The aim of this paper is to identify the geometric and algebraic structures associated to discrete memoryless channels. The procedure employed to achieve this goal is based on the following steps: knowing the graph associated to the discrete memoryless channel, 1) determine the set of surfaces in which the graph is embedded; 2) establish the set of algebraic structures inherited by the surfaces through the first homology group; and 3) identify the regular tessellations which may be used in the design of modulators and quantizers.

We investigate a method of communicating with chaos described by Hayes, Grebogi, and Ott (19913), which uses the idea of controlling chaos by small perturbations. Although allowing the electronics controlling the output signal to remain at the low-power level, this method unavoidably incurs a capacity loss, which we calculate for a chaotic CO/sub 2/ laser model. We first determine an approximation to the laser language, by means of a forbidden set representation, and then find its Shannon (1963) capacity. The results indicate that, since the capacity loss is not significant, the tradeoff is worthwhile. Finally, we point to simple finite-state encoders that satisfy the constraints imposed by the language.

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric module a two-dimensional (2-D) grid, in particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

We present a decoding procedure for Reed-Solomon (RS) and BCH codes defined over an integer residue ring pgZ/sub q/, where q is a power of a prime p. The proposed decoding procedure, as for RS and BCH codes over fields, consists of four major steps: (1) calculation of the syndromes; (2) calculation of the "elementary symmetric functions," by a modified Berlekamp-Massey (1968, 1969) algorithm for commutative rings; (3) calculation of the error location numbers; and (4) calculation of the error magnitudes. The proposed decoding procedure also applies to the synthesis of a shortest linear-feedback shift register (LFSR), capable of generating a prescribed finite sequence of elements lying in a commutative ring with identity.

It is known that group block codes over nonabelian groups of order less than 32 are asymptotically bad. In this correspondence we extend that result, showing that group block codes over nonabelian groups are asymptotically bad, no matter what the order of the group under consideration is.

We present several new ternary and quaternary codes found with a heuristic search procedure, originally proposed by the authors to find good binary codes, and which is based on combinatorial optimization.

We propose a technique that can be used to realize the optimum allocation of frequencies for a mobile radio telephone system and for multibeam satellites by employing the quadratic forms associated with the rectangular and the hexagonal grids.

Presents an alternative way of looking at convolutional codes as a network of flows. From this association, previous known classes of nonsystematic convolutional codes are reproduced. The authors also present a decomposition of the problem of finding good convolutional codes into well known and structured problems such as the maximum flow problem, the knapsack problem, and the modular representation of block codes. A spanning tree search algorithm is also proposed for finding good codes. The authors consider the class of nonsystematic convolutional codes for which the encoders satisfy an equivalent property related to the feasibility of flow. Finally, extensive tables show the new and the previous known codes.>

A multistage decoding algorithm is given for lattices obtained from a multilevel code formula. The algorithm is shown to have the same effective error-correcting radius as maximum-likelihood decoding, so that the performance loss is essentially determined by the increase in the effective error coefficient, for which an expression is given. The code formula generalizes some previous multilevel constructions to constructions of known single-level binary lattices with many levels, and then to decoders for them with the proposed algorithm. The trade-off between complexity reduction and performance loss is evaluated for several known lattices and two new ones, indicating that the approach is effective provided the binary codes involved in the code formula are not too short. All codes used in our constructions are binary Reed-Muller codes.>

A periodically-time-varying binary (3, 2) convolutional code with period T=2 and memory M=1 having larger free distance than the best time-invariant (3, 2) convolutional code with M=1 is presented.>

A method for designing good unit-memory convolutional codes is presented. The method is based on the decomposition of the original problem into two easier subproblems that can be formulated as optimization problems and solved by efficient heuristic search algorithms. The efficacy of this method is demonstrated by a table containing 33 new unit-memory convolutional codes (n,k) with 5>

The authors propose a decoding strategy based on the multistage decoding method of Imai and Hirakawa (1977). This scheme is shown to be a generalization of the conventional multistage decoders. A general procedure to derive an upper bound on the bit error probability, which takes into account wrong side information effects, is stated. Finally. a comparison with conventional multistage decoders is presented to show that the proposed decoding strategy outperforms the previous ones.>