On the Error-Correcting Performance of some Binary and Ternary Linear Codes

In this work, we determine the coset weight spectra of all binary cyclic codes of lengths up to 33, ternary cyclic and negacyclic codes of lengths up to 20 and of some binary linear codes of lengths up to 33 which are distance-optimal, by using some of the algebraic properties of the codes and a computer assisted search. Having these weight spectra the monotony of the function of the undetected error probability after t-error correction P(t)ue (C,p) could be checked with any precision for a linear time. We have used a programm written in Maple to check the monotony of P(t)ue (C,p) for the investigated codes for a finite set of points of p € [0, p/(q-1)] and in this way to determine which of them are not proper.

Improving the Watermarking Process with Usage of Block Error-Correcting Codes

The emergence of digital imaging and of digital networks has made duplication of original artwork easier. Watermarking techniques, also referred to as digital signature, sign images by introducing changes that are imperceptible to the human eye but easily recoverable by a computer program. Usage of error correcting codes is one of the good choices in order to correct possible errors when extracting the signature. In this paper, we present a scheme of error correction based on a combination of Reed-Solomon codes and another optimal linear code as inner code. We have investigated the strength of the noise that this scheme is steady to for a fixed capacity of the image and various lengths of the signature. Finally, we compare our results with other error correcting techniques that are used in watermarking. We have also created a computer program for image watermarking that uses the newly presented scheme for error correction.

Experiments in Detection and Correction of Russian Malapropisms by Means of the WEB

Malapropism is a semantic error that is hardly detectable because it usually retains syntactical links between words in the sentence but replaces one content word by a similar word with quite different meaning. A method of automatic detection of malapropisms is described, based on Web statistics and a specially defined Semantic Compatibility Index (SCI). For correction of the detected errors, special dictionaries and heuristic rules are proposed, which retains only a few highly SCI-ranked correction candidates for the user’s selection. Experiments on Web-assisted detection and correction of Russian malapropisms are reported, demonstrating efficacy of the described method.

Paronyms for Accelerated Correction of Semantic Errors

* Work done under partial support of Mexican Government (CONACyT, SNI), IPN (CGPI, COFAA) and Korean Government (KIPA Professorship for Visiting Faculty Positions). The second author is currently on Sabbatical leave at Chung-Ang University.

On Graph-Based Cryptography and Symbolic Computations

We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs.

On Finite Element Methods for 2nd order (semi–) periodic Eigenvalue Problems

We deal with a class of elliptic eigenvalue problems (EVPs) on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions (BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational formulation which is shown to fit into the general framework of abstract EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs) without and with numerical quadrature. The aim of the paper is to show that well–known error estimates, established for the finite element approximation of elliptic EVPs with classical BCs, hold for the present types of EVPs too. Some attention is also paid to the computational aspects of the resulting algebraic EVP. Finally, the analysis is illustrated by two non-trivial numerical examples, the exact eigenpairs of which can be determined.

Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration provide a convergence rate of O(N^(−1/2)) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)^k)N^(−1)). We have shown theoretically and through numerical tests that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.