Abstract
We classify all self-reciprocal polynomials arising from reversed Dickson polynomials over \(\mathbb {Z}\) and \(\mathbb {F}_p\), where p is prime. As a consequence, we also obtain coterm polynomials arising from reversed Dickson polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abualrub T., Ghrayeb A., Zeng X.N.: Construction of cyclic codes over \(GF(4)\) for DNA computing. WSEAS Trans. Math. 5(6), 750–755 (2006).
Adleman L.: Molecular computation of solutions to combinatorial problems. Science 266, 1021–1024 (1994).
Fernando N.: Reversed Dickson polynomials of the \((k+1)\)-th kind over finite fields. J. Number Theory 172, 234–255 (2017).
Fernando N.: Reversed Dickson polynomials of the third kind. arXiv:1602.04545.
Guenda K., Gulliver T.A.: Construction of cyclic codes over \(\mathbb{F}_2+u\mathbb{F}_2\) for DNA computing. Appl. Algebra Eng. Commun. Comput. 24(6), 445–459 (2013).
Guenda K., Gulliver T.A., Solé P.: On cyclic DNA codes. In: Proc. IEEE Int. Symp. Inform. Theory, pp. 121–125, Istanbul (2013).
Guenda K., Jitman S., Gulliver T.A.: Constructions of good entanglement-assisted quantum error correcting codes. arXiv:1606.00134.
Hoffman D.G., Leonard D.A., Lindner C.C., Phelps K.T., Rodger C.A., Wall J.R.: Coding theory. The essentials. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 150. Marcel Dekker, Inc., New York (1991).
Hong S.J., Bossen D.C.: On some properties of self-reciprocal polynomials. IEEE Trans. Inf. Theory 21, 462–464 (1975).
Hong S., Qin X., Zhao W.: Necessary conditions for reversed Dickson polynomials of the second kind to be permutational. Finite Fields Appl. 37, 54–71 (2016).
Hou X., Ly T.: Necessary conditions for reversed Dickson polynomials to be permutational. Finite Fields Appl. 16, 436–448 (2010).
Massey J.L.: Reversible codes. Inf. Control 7, 369–380 (1964).
Meyn H.: On the construction of irreducible self-reciprocal polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 1, 43–53 (1990).
Miller R.L.: Necklaces, symmetries and self-reciprocal polynomials. Discret. Math. 22, 25–33 (1978).
Oztas E.S., Siap I., Yildiz B.: Reversible codes and applications to DNA. In: Lecture Notes in Computer Science, vol. 8592. Springer, Heidelberg (2014).
Pattanayak S., Singh A.K.: Construction of cyclic DNA codes over the Ring \(\mathbb{Z}_4[u]/\langle u^2-1 \rangle \) based on the deletion distance. arXiv:1603.04055.
Pintoptang U., Laohakosol V., Tadee S.: Necklaces, self-reciprocal polynomials, and \(q\)-cycles. Int. J. Comb. (2014), Art. ID 593749.
Pless V.: Introduction to the theory of error-correcting codes. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication, 3rd edn. Wiley, New York (1998).
Siap I., Abualrub T., Ghrayeb A.: Cyclic DNA codes over the Ring \(\mathbb{F}_2[u]/\langle u^2-1 \rangle \) based on the deletion distance. J. Franklin Inst. 346(8), 731–740 (2009).
Yang X., Massey J.L.: The condition for a cyclic code to have a complementary dual. Discret. Math. 126(1–3), 391–393 (1994).
Yildiz B., Siap I.: Cyclic codes over \(\mathbb{F}_2[u]/\langle u^4-1 \rangle \) and applications to DNA codes. Comput. Math. Appl. 63(7), 1169–1176 (2012).
Yucas J., Mullen G.: Self-reciprocal irreducible polynomials over finite fields. Des. Codes Cryptogr. 33(3), 275–281 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Mullen.
Rights and permissions
About this article
Cite this article
Fernando, N. Self-reciprocal polynomials and coterm polynomials. Des. Codes Cryptogr. 86, 1707–1726 (2018). https://doi.org/10.1007/s10623-017-0419-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-017-0419-4