# Igor Gachkov

## Forskning

Applications of the theories of Error-Correcting Codes have increased tremendously in recent years. Thus it is hardly possible today to imagine engineers working with data transmission and related fields without basic knowledge of coding/decoding of information. The possibilities of quantifying information with electronic equipment are developing rapidly, supplying the specialists working in communication theory with more sophisticated methods for circuit realization of concrete algorithms in Coding Theory.The most part of my research is discrete mathematics and computer algebra as well as theory of error-correcting codes. It is well known that unknown perfect codes do not exist. Their parameters are completely equivalent with parameters of Hamming and Golay codes. (A.Tietäväinen On the nonexisterance of perfect codes over finite fields, SIAM J.Appl.Math 24 (1973) ). The most important and interesting areas in coding theory are development of new methods for building of quasi-perfect codes. There are a short number of such codes. Among all known examples an error-correcting BCH-code can be named. Every new discovery of a quasi-perfect is a big achievement in the area. I have constructed a new type of quasi-perfect triple code. ( ”Linear triple quasi-perfect codes”, Probl. pered.inf. XXII,4, (1986). As further research I studied RS-codes (Reed-Solomon codes) over GF(8) and their binary limit. A well-chosen basis provides an opportunity to increase the code distance. I suggest a simple method to build quasi-perfect Wagner's codes with parameters [23,14,5] (Wagner T. J. A search technique for quasi-perfect codes Info. and Control, 9 (1966) 94-99), the weight polynom of the code was evaluated too. I formed a new class of codes based on linear fraction functions with coefficients from the limited body Fq. This class is wider in comparison with Goppa codes. Those codes are described in details including parameters and automorfism groups. Limits of the codes at the binary body give a way to build new quasi-perfect codes.

## Utvalda publikationer

Some Remarks on Testing Irreducibility of Polynomials and Normality of Bases in Finite Fields, Fundamenta Informaticae 104 (2010) pp. 227-238

Bit parallel circuits for arithmetic operations in composite fields $ GF(2^{nm}) (CMMSE) Proceedings ISBN: 978-84-613-5510-5 pp. 296-306

Probabilistic algorithm to find a normal basis in special finite fields. (CMMSE) Proceedings ISBN: 978-84-612-9727-6 pp. 532-536

Using the package “Coding Theory” for a search technique for Quasi-perfect Codes. International e-Conference of Computer Science. (IeCCS) AIP Conference Proceedings 2008, vol 1060 pp.133-137 AIP American Institute of Physics

Visualisation of the Mathematical Process: Boolean Algebra and Graph Theory with TI-83/89. Journal of the Korea Society of Mathematical Education Vol.11, No.2, pp. 143-151

New Optimal Constant Weight Codes The electronic journal of combinatorics 14 , no. 1, Note 13, pp.1-6 7.(2007) ( i samarbete med Henrik Larsson) Improvements on the Juxtaposing theorem SerdicaJ.Computing1 Volume 1, Number 2, pp.207-212

A geometric approach to finding new lower bounds of A ( n , d , w ) , Designs, Codes and Cryptography v. 43, N 2-3 / June, 2007 pp. 85-91

Optimal Constant Weight Codes Lecture note in Computer science LNCS 3991 pp. 912 – 915

## Publikationer

- S. B. Gashkov, Igor Gashkov, 2018
- Igor Gachkov, A.A Burtsev, R.A Khokhlov, I Gashkov, 2010
- Igor Gachkov, S.B Gashkov, 2010
- Igor Gachkov, S.B Gashkov, 2009
- Igor Gachkov, 2008
- Igor Gachkov, A.O. Ekberg, D. Taub, 2007
- Igor Gachkov, H Larsson, 2007
- Igor Gachkov, D Taub, 2007
- Igor Gachkov, 2007
- S. Gashkov, Igor Gachkov, 2006
- Igor Gachkov, 2006
- A. Burtsev, S. Gashkov, Igor Gachkov, 2006
- S. Gashkov, Igor Gachkov, 2006
- Igor Gachkov, 2006
- S Gashkov, Igor Gachkov, 2005
- Igor Gachkov, 2003
- Igor Gachkov, Kazarin , Sidelnikov , 2003