| Title:
|
On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan (English) |
| Author:
|
Ugolini, Simone |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
243-250 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
finite field |
| Keyword:
|
irreducible polynomial |
| Keyword:
|
iterative construction |
| MSC:
|
11R09 |
| MSC:
|
11T55 |
| MSC:
|
12E05 |
| idZBL:
|
Zbl 06587887 |
| idMR:
|
MR3483236 |
| DOI:
|
10.1007/s10587-016-0253-2 |
| . |
| Date available:
|
2016-04-07T15:10:36Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144889 |
| . |
| Reference:
|
[1] Cohen, S. D.: The explicit construction of irreducible polynomials over finite fields.Des. Codes Cryptography 2 (1992), 169-174. MR 1171532, 10.1007/BF00124895 |
| Reference:
|
[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 |
| Reference:
|
[3] Kyuregyan, M. K.: Iterated constructions of irreducible polynomials over finite fields with linearly independent roots.Finite Fields Appl. 10 (2004), 323-341. Zbl 1049.11135, MR 2067602 |
| Reference:
|
[4] Kyuregyan, M. K.: Recurrent methods for constructing irreducible polynomials over {$ GF(2)$}.Finite Fields Appl. 8 (2002), 52-68. Zbl 1028.11073, MR 1872791 |
| Reference:
|
[5] Meyn, H.: On the construction of irreducible self-reciprocal polynomials over finite fields.Appl. Algebra Eng. Commun. Comput. 1 (1990), 43-53. MR 1325510, 10.1007/BF01810846 |
| Reference:
|
[6] Mullen, G. L., Panario, D.: Handbook of Finite Fields.Discrete Mathematics and Its Applications CRC Press, Boca Raton (2013). Zbl 1319.11001, MR 3087321 |
| Reference:
|
[7] Ugolini, S.: Sequences of irreducible polynomials without prescribed coefficients over odd prime fields.Des. Codes Cryptography 75 (2015), 145-155. Zbl 1319.11071, MR 3320357, 10.1007/s10623-013-9897-1 |
| Reference:
|
[8] Ugolini, S.: Sequences of binary irreducible polynomials.Discrete Math. 313 (2013), 2656-2662. Zbl 1283.11161, MR 3095441, 10.1016/j.disc.2013.08.011 |
| Reference:
|
[9] Ugolini, S.: Graphs associated with the map {$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. Zbl 1302.37074, MR 2975769 |
| Reference:
|
[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 |
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