Recursive Polynomial Sets and Their Some Algebraic Applications
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
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.