Previous |  Up |  Next


MSC: 11R09, 11T55, 12E05
Full entry | Fulltext not available (moving wall 24 months)      Feedback
finite field; irreducible polynomial; iterative construction
We deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the $Q$-transform employed previously by Varshamov and Garakov (1969) as well as by Meyn (1990) for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper these conditions are removed. We construct infinite sequences of irreducible polynomials of non-decreasing degree starting from any irreducible polynomial.
[1] Cohen, S. D.: The explicit construction of irreducible polynomials over finite fields. Des. Codes Cryptography 2 (1992), 169-174. DOI 10.1007/BF00124895 | MR 1171532
[2] Green, D. H., Taylor, I. S.: Irreducible polynomials over composite Galois fields and their applications in coding techniques. Proc. Inst. Elec. Engrs. 121 (1974), 935-939. MR 0434611
[3] Kyuregyan, M. K.: Iterated constructions of irreducible polynomials over finite fields with linearly independent roots. Finite Fields Appl. 10 (2004), 323-341. MR 2067602
[4] Kyuregyan, M. K.: Recurrent methods for constructing irreducible polynomials over {$ GF(2)$}. Finite Fields Appl. 8 (2002), 52-68. MR 1872791
[5] Meyn, H.: On the construction of irreducible self-reciprocal polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 1 (1990), 43-53. DOI 10.1007/BF01810846 | MR 1325510
[6] Mullen, G. L., Panario, D.: Handbook of Finite Fields. Discrete Mathematics and Its Applications CRC Press, Boca Raton (2013). MR 3087321
[7] Ugolini, S.: Sequences of irreducible polynomials without prescribed coefficients over odd prime fields. Des. Codes Cryptography 75 (2015), 145-155. DOI 10.1007/s10623-013-9897-1 | MR 3320357
[8] Ugolini, S.: Sequences of binary irreducible polynomials. Discrete Math. 313 (2013), 2656-2662. DOI 10.1016/j.disc.2013.08.011 | MR 3095441
[9] Ugolini, S.: Graphs associated with the map {$x\mapsto x+x^{-1}$} in finite fields of characteristic two. Theory and Applications of Finite Fields. Conf. on finite fields and their applications, Ghent, Belgium, 2011 American Mathematical Society, Contemporary Mathematics 579 Providence (2012), 187-204 M. Lavrauw et al. MR 2975769
[10] Varšamov, R. R., Garakov, G. A.: On the theory of selfdual polynomials over a Galois field. Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 13 Russian (1969), 403-415. MR 0297454
Partner of
EuDML logo