Sergey Bezzateev

Decoding Cyclic Codes up to a New Bound on the Minimum Distance

A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem bound and, for some codes, upon the Hartmann-Tzeng bound. Several Boston bounds are special cases of our bound. For some classes of codes, the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forneys formula.

Securing Network-Assisted Direct Communication: The Case of Unreliable Cellular Connectivity

Network-assisted device-to-device (D2D) communication is a next-generation wireless technology enabling direct connectivity between proximate user devices under the control of cellular infrastructure. It couples together the centralized and the distributed network architectures, and as such requires respective enablers for secure, private, and trusted data exchange especially when cellular control link is not available at all times. In this work, we conduct the state-of-the-art overview and propose a novel algorithm to maintain security functions of proximate devices in case of unreliable cellular connectivity, whether a new device joins the secure group of users or an existing device leaves it. Our proposed solution and its rigorous mathematical implementation detailed in this work open door to a novel generation of secure proximity-based services and applications in future wireless communication systems.

A New Bound on the Minimum Distance of Cyclic Codes Using Small-Minimum-Distance Cyclic Codes

A new bound on the minimum distance of q-ary cyclic codes is proposed. It is based on the description by another cyclic code with small minimum distance. The connection to the BCH bound and the Hartmann–Tzeng (HT) bound is formulated explicitly. We show that for many cases our approach improves the HT bound. Furthermore, we refine our bound for several families of cyclic codes. We define syndromes and formulate a Key Equation that allows an efficient decoding up to our bound with the Extended Euclidean Algorithm. It turns out that lowest-code-rate cyclic codes with small minimum distances are useful for our approach. Therefore, we give a sufficient condition for binary cyclic codes of arbitrary length to have minimum distance two or three and lowest code-rate.

Class of generalized Goppa codes perfect in weighted Hamming metric

A weighted Hamming metric is considered. A class of binary linear codes consistent with the weighted Hamming metric is discussed. A class of binary generalized Goppa codes perfect in the weighted Hamming metric is offered.

Subclass of Cyclic Goppa Codes

The subclass of cyclic Goppa codes is proposed. It is proved that this subclass contains all cyclic codes of length n|q^m±1 with generator polynomial g(x), g(αⁱ)=g(α^-j)=0, i=0,1,..., s₁, j=1,..., s₂, αⁿ=1, α ∈ GF(q^2m).

Steganographic method on weighted container

This paper discusses the approach to construction of steganographic method for the weighted container. To find the efficient solution, concept of Penalty Function is introduced. The analysis of the efficiency of the weighted embedding for the simple model of weighted container is shown.

Chain of Separable Binary Goppa Codes and Their Minimal Distance

It is shown that subclasses of separable binary Goppa codes, <i>Gamma(L</i>,<i>G</i>) - codes, with <i>L</i>={alpha isin GF(2^2l):<i>G</i>(alpha) ne 0 } and special Goppa polynomials <i>G</i>(<i>x</i>) can be presented as a chain of embedded and equivalent codes. The true minimal distance has been obtained for all codes of the chain.

Generalizing bounds on the minimum distance of cyclic codes using cyclic product codes

Two generalizations of the Hartmann-Tzeng (HT) bound on the minimum distance of q-ary cyclic codes are proposed. The first one is proven by embedding the given cyclic code into a cyclic product code. Furthermore, we show that unique decoding up to this bound is always possible and outline a quadratic-time syndrome-based error decoding algorithm. The second bound is stronger and the proof is more involved. Our technique of embedding the code into a cyclic product code can be applied to other bounds, too and therefore generalizes them.

Joint safety and security analysis for complex systems

The problem of joint safety and security analysis is considered. For complex systems method of fault tree analysis for safety and security is proposed. The effectiveness of new approach of joint safety and security analysis is shown on example of the European Railway Traffic Management System (ERTMS).

Efficient decoding of some classes of binary cyclic codes beyond the Hartmann-Tzeng bound

A new bound on the distance of binary cyclic codes is proposed. The approach is based on the representation of a subset of the roots of the generator polynomial by a rational function. A new bound on the minimum distance is proven and several classes of binary cyclic codes are identified. For some classes of codes, this bound is better than the known bounds (e.g. BCH or Hartmann-Tzeng bound). Furthermore, a quadratic-time decoding algorithm up to this new bound is developed.