Research Article
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Recursive Polynomial Sets and Their Some Algebraic Applications

Year 2025, Volume: 46 Issue: 2, 405 - 409, 30.06.2025

Hacı Aktaş

https://doi.org/10.17776/csj.1524109

Abstract

This paper primarily defines the framework for a new class of polynomial sets over a finite field GF (2), providing a recursive definition and delving into pertinent algebraic properties. We also studied some applications of the obtained polynomial classes on coding theory, such as obtaining new code classes. Our focus lies on polynomial sets with degrees equal to or less than n, for which we present a methodology for encoding and decoding utilizing an irreducible polynomial p(x) = xm+xs+1, (m = 2n-1). Furthermore, as an application of this method in coding theory, we created new code classes and studied some features of these codes.

Keywords

Polynomial sets Polynomial codes generator matrix

References

  • [1] Cadilhac M., Mazowiecki F., Paperman C., Pilipczuk M., S´enizergues G., On Polynomial Recursive Sequences, Theory of Computing Systems, (68) (2024) 593–614.
  • [2] Fu S., Fenga X., Linc D., Wangd Q., A Recursive Construction of Permutation Polynomials over F_(q^2 ) with OddCharacteristic from R ́edei Functions, Designs, Codes and Cryptography, 87 (2019) 1481–1498.
  • [3] Sidki S., Sadaka R., Benazzouz A., Computing recursive orthogonal polynomial with Schur complements, Journal of Computational and Applied Mathematics, 373 (2020) 112406.
  • [4] Shannon C., A Mathematical Theory of Communication, The Bell System Technical Journal, 27 (1948) 379–423, 623–656.
  • [5] Dinga C., Ling S., Aq-polynomial approach to cyclic codes, Finite Fields and Their Applications, 20 (2013) 1-14.
  • [6] R. Abdullaev, D. Efanov, Polynomial Codes Properties Application in Concurrent Error-Detection Systems of Combinational Logic Devices, IEEE East-West Design & Test Symposium (EWDTS), Batumi, Georgia, 2021.
  • [7] Mao-Ching Chiu, Polynomial Representations of Polar Codes and Decoding under Overcomplete Representations, IEEE Communications Letters, 17 (12) (2013) 2340-2343.
  • [8] Wang X., Hao Y., Qiao D., Constructions of Polynomial Codes Based on Circular Permutation Over Finite Fields, IEEE, 8 (2020) 134219 - 134223.
  • [9] Nalli A. and Haukkanen P., On generalized Fibonacci and Lucas polynomials, Chaos Solitons And Fractals, 42 (2009) 3179-3186.
  • [10] Prasad B., Coding theory on (h(x), g(y))-extension of Fibonacci p-numbers polynomials, Universal Journal of Computational Mathematics, 2 (1) (2014) 6-10.
  • [11] Stakhov A. P., Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos Solitons and Fractals, 30 (1) (2006) 56-66.
  • [12] Kaymak O. O., Coding theory for h(x)-Fibonacci polynomials, J. BAUN Inst. Sci. Technol., 26 (1) (2024) 226-236
  • [13] Hill R., A First Course in Coding Theory. Oxford: Clarendon Press, (1986) 1-67.
  • [14] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes. New York: North-holland Publishing company, (1977) 1-37, 188-215.
  • [15] Ling S., Xing C., Coding Theory A First Course. Cambridge University Press, (2004) 39-57.
  • [16] Hoffman D. G., Leonard D. A., Lindner C. C., Phelps K. T., Rodger C. A., Wall J. R., Coding Theory, Marcel Dekker Inc., New York, (1991)29-117.Alam M. N., Bonyah E., Fayz-Al-Asad M., Reliable analysis for the Drinfeld-Sokolov-Wilson equation in mathematical physics, Palest. J. Math., 11 (1) (2022) 397-407.

Year 2025, Volume: 46 Issue: 2, 405 - 409, 30.06.2025

Hacı Aktaş

https://doi.org/10.17776/csj.1524109

Abstract

References

  • [1] Cadilhac M., Mazowiecki F., Paperman C., Pilipczuk M., S´enizergues G., On Polynomial Recursive Sequences, Theory of Computing Systems, (68) (2024) 593–614.
  • [2] Fu S., Fenga X., Linc D., Wangd Q., A Recursive Construction of Permutation Polynomials over F_(q^2 ) with OddCharacteristic from R ́edei Functions, Designs, Codes and Cryptography, 87 (2019) 1481–1498.
  • [3] Sidki S., Sadaka R., Benazzouz A., Computing recursive orthogonal polynomial with Schur complements, Journal of Computational and Applied Mathematics, 373 (2020) 112406.
  • [4] Shannon C., A Mathematical Theory of Communication, The Bell System Technical Journal, 27 (1948) 379–423, 623–656.
  • [5] Dinga C., Ling S., Aq-polynomial approach to cyclic codes, Finite Fields and Their Applications, 20 (2013) 1-14.
  • [6] R. Abdullaev, D. Efanov, Polynomial Codes Properties Application in Concurrent Error-Detection Systems of Combinational Logic Devices, IEEE East-West Design & Test Symposium (EWDTS), Batumi, Georgia, 2021.
  • [7] Mao-Ching Chiu, Polynomial Representations of Polar Codes and Decoding under Overcomplete Representations, IEEE Communications Letters, 17 (12) (2013) 2340-2343.
  • [8] Wang X., Hao Y., Qiao D., Constructions of Polynomial Codes Based on Circular Permutation Over Finite Fields, IEEE, 8 (2020) 134219 - 134223.
  • [9] Nalli A. and Haukkanen P., On generalized Fibonacci and Lucas polynomials, Chaos Solitons And Fractals, 42 (2009) 3179-3186.
  • [10] Prasad B., Coding theory on (h(x), g(y))-extension of Fibonacci p-numbers polynomials, Universal Journal of Computational Mathematics, 2 (1) (2014) 6-10.
  • [11] Stakhov A. P., Fibonacci matrices, a generalization of the Cassini formula and a new coding theory, Chaos Solitons and Fractals, 30 (1) (2006) 56-66.
  • [12] Kaymak O. O., Coding theory for h(x)-Fibonacci polynomials, J. BAUN Inst. Sci. Technol., 26 (1) (2024) 226-236
  • [13] Hill R., A First Course in Coding Theory. Oxford: Clarendon Press, (1986) 1-67.
  • [14] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes. New York: North-holland Publishing company, (1977) 1-37, 188-215.
  • [15] Ling S., Xing C., Coding Theory A First Course. Cambridge University Press, (2004) 39-57.
  • [16] Hoffman D. G., Leonard D. A., Lindner C. C., Phelps K. T., Rodger C. A., Wall J. R., Coding Theory, Marcel Dekker Inc., New York, (1991)29-117.Alam M. N., Bonyah E., Fayz-Al-Asad M., Reliable analysis for the Drinfeld-Sokolov-Wilson equation in mathematical physics, Palest. J. Math., 11 (1) (2022) 397-407.
There are 16 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Natural Sciences
Authors

Hacı Aktaş 0009-0001-6977-5133

Publication Date June 30, 2025
Submission Date July 30, 2024
Acceptance Date March 12, 2025
Published in Issue Year 2025Volume: 46 Issue: 2

Cite

APA Aktaş, H. (2025). Recursive Polynomial Sets and Their Some Algebraic Applications. Cumhuriyet Science Journal, 46(2), 405-409. https://doi.org/10.17776/csj.1524109