On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

Simone Ugolini

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 243-250
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Ugolini, Simone. "On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan." Czechoslovak Mathematical Journal 66.1 (2016): 243-250. <http://eudml.org/doc/276789>.

@article{Ugolini2016,
abstract = {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.},
author = {Ugolini, Simone},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite field; irreducible polynomial; iterative construction},
language = {eng},
number = {1},
pages = {243-250},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan},
url = {http://eudml.org/doc/276789},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Ugolini, Simone
TI - On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 243
EP - 250
AB - 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.
LA - eng
KW - finite field; irreducible polynomial; iterative construction
UR - http://eudml.org/doc/276789
ER -

References

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  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. (1974) MR0434611
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  4. Kyuregyan, M. K., Recurrent methods for constructing irreducible polynomials over G F ( 2 ) , Finite Fields Appl. 8 (2002), 52-68. (2002) Zbl1028.11073MR1872791
  5. Meyn, H., 10.1007/BF01810846, Appl. Algebra Eng. Commun. Comput. 1 (1990), 43-53. (1990) MR1325510DOI10.1007/BF01810846
  6. Mullen, G. L., Panario, D., Handbook of Finite Fields, Discrete Mathematics and Its Applications CRC Press, Boca Raton (2013). (2013) Zbl1319.11001MR3087321
  7. Ugolini, S., 10.1007/s10623-013-9897-1, Des. Codes Cryptography 75 (2015), 145-155. (2015) Zbl1319.11071MR3320357DOI10.1007/s10623-013-9897-1
  8. Ugolini, S., 10.1016/j.disc.2013.08.011, Discrete Math. 313 (2013), 2656-2662. (2013) Zbl1283.11161MR3095441DOI10.1016/j.disc.2013.08.011
  9. Ugolini, S., Graphs associated with the map x ↦ 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. (2012) Zbl1302.37074MR2975769
  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. (1969) MR0297454

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