Abstract
In this note we have revisited some of the results of Trachtenberg (On the cross-correlation functions of maximal linear sequences, Ph.D. thesis, University of Southern California, Los Angeles, 1970), which are directly related with the number of solutions of some special linearized polynomials over finite fields. In some cases we give improvements. Also, we give some results on the exact number of solutions of certain linearized equations depending on the coefficients of that equation.
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References
Bluher A.W.: On \(x^{q+1}+ax+b\). Finite Fields Appl. 10(3), 285–305 (2004).
Choi S.-T., No J.-S., Chung H.: On the cross-correlation of a ternary \(m\)-sequence of period \(3^{4k+2} - 1\) and its decimated sequence by \(\frac{(3^{2k+1}+1)^2}{8}\). In: Proceedings of the IEEE International Symposium on Information Theory, pp. 1268–1271 (2010).
Choi S.-T., Lim T.-H., No J.-S., Chung H.: Evaluation of cross-correlation values of \(p\)-ary \(m\)-sequence and its decimated sequence by \(\frac{p^n+1}{p+1} + \frac{p^n-1}{2}\). In: Proceedings of the IEEE International Symposium on Information Theory, pp. 683–687 (2011).
Choi S.-T., Lim T.-H., No J.-S., Chung H.: On the cross-correlation of a \(p\)-Ary \(m\)-sequence of period \(p^{2m}-1\) and its decimated sequences by \((p^m+1)^2/2(p+1)\). IEEE Trans. Inf. Theory 58(3), 1873–1879 (2012).
Helleseth T., Gong G.: New nonbinary sequences with ideal two-level autocorrelation. IEEE Trans. Inf. Theory 48(11), 2868–2872 (2002).
Helleseth T., Gong G., Hu H.: A three-valued Walsh transform from decimations of Helleseth-Gong sequences. IEEE Trans. Inf. Theory 58(2), 1158–1162 (2012).
Helleseth T., Gong G., Hu H., Li C.: New three-valued Walsh transforms from decimations of Helleseth-Gong sequences. In: Sequences and Their Applications, SETA 2012. LNCS 7280, pp. 327–337. Springer, Heidelberg (2012).
Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).
Luo J., Helleseth T., Kholosha A.: Two nonbinary sequences with six-valued cross correlation. In: International Workshop on Signal Design and Its Applications in Communications, IWSDA 2011, pp. 44–47 (2011).
Trachtenberg H.M.: On the cross-correlation functions of maximal linear sequences. Ph.D. thesis, University of Southern California, Los Angeles (1970).
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments. The work of Z. Saygı was supported by TÜBİTAK under Grant No. TBAG-109T344.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Özbudak, F., Saygı, Z. On the exact number of solutions of certain linearized equations. Des. Codes Cryptogr. 73, 457–468 (2014). https://doi.org/10.1007/s10623-014-9942-8
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DOI: https://doi.org/10.1007/s10623-014-9942-8