# American Institute of Mathematical Sciences

May  2016, 10(2): 401-411. doi: 10.3934/amc.2016014

## Nearly perfect sequences with arbitrary out-of-phase autocorrelation

 1 Department of Mathematics, Hacettepe University, Beytepe, 06800, Ankara

Received  September 2014 Published  April 2016

A sequence of period $n$ is called a nearly perfect sequence of type $\gamma$ if all out-of-phase autocorrelation coefficients are a constant $\gamma$. In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a $p$-ary NPS of period $n$ and type $\gamma$ and a cyclic $(n,p,n,\frac{n-\gamma}{p}+\gamma,0,\frac{n-\gamma}{p})$-DPDS for an arbitrary integer $\gamma$. Next, we present the necessary conditions for the existence of a $p$-ary NPS of type $\gamma$. We apply this result for excluding the existence of some $p$-ary NPS of period $n$ and type $\gamma$ for $n \leq 100$ and $\vert \gamma \vert \leq 2$. We also prove the similar results for an almost $p$-ary NPS of type $\gamma$. Finally, we show the non-existence of some almost $p$-ary perfect sequences by showing the non-existence of equivalent cyclic relative difference sets by using the notion of multipliers.
Citation: Oǧuz Yayla. Nearly perfect sequences with arbitrary out-of-phase autocorrelation. Advances in Mathematics of Communications, 2016, 10 (2) : 401-411. doi: 10.3934/amc.2016014
##### References:
 [1] T. Beth, D. Jungnickel and H. Lenz, Design Theory,, 2nd edition, (1999).   Google Scholar [2] B. W. Brock, Hermitian congruence and the existence and completion of generalized Hadamard matrices,, J. Combin. Theory Ser. A, 49 (1988), 233.  doi: 10.1016/0097-3165(88)90054-4.  Google Scholar [3] Y. M. Chee, Y. Tan and Y. Zhou, Almost $p$-ary perfect sequences,, in Sequences and their Applications - SETA 2010, (2010), 399.  doi: 10.1007/978-3-642-15874-2_34.  Google Scholar [4] T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory, ().   Google Scholar [5] D. Jungnickel and A. Pott, Perfect and almost perfect sequences,, Discrete Appl. Math., 95 (1999), 331.  doi: 10.1016/S0166-218X(99)00085-2.  Google Scholar [6] S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences,, Int. J. Inf. Coding Theory, 1 (2009), 15.  doi: 10.1504/IJICOT.2009.024045.  Google Scholar [7] S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices,, J. Algebra, 175 (1995), 505.  doi: 10.1006/jabr.1995.1198.  Google Scholar [8] S. L. Ma and B. Schmidt, On $(p^a,p,p^a,p^{a-1})$-relative difference sets,, Des. Codes Crypt., 6 (1995), 57.  doi: 10.1007/BF01390771.  Google Scholar [9] F. Özbudak, O. Yayla and C. C. Yíldírím, Nonexistence of certain almost $p$-ary perfect sequences,, in Sequences and their Applications - SETA 2012, (2012), 13.  doi: 10.1007/978-3-642-30615-0_2.  Google Scholar [10] A. Pott, Finite Geometry and Character Theory,, Springer-Verlag, (1995).   Google Scholar [11] R. J. Turyn, Character sums and difference sets,, Pacific J. Math., 15 (1965), 319.   Google Scholar [12] A. Winterhof, O. Yayla and V. Ziegler, Non-existence of some nearly perfect sequences, near Butson-Hadamard matrices, and near conference matrices,, preprint, ().   Google Scholar

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##### References:
 [1] T. Beth, D. Jungnickel and H. Lenz, Design Theory,, 2nd edition, (1999).   Google Scholar [2] B. W. Brock, Hermitian congruence and the existence and completion of generalized Hadamard matrices,, J. Combin. Theory Ser. A, 49 (1988), 233.  doi: 10.1016/0097-3165(88)90054-4.  Google Scholar [3] Y. M. Chee, Y. Tan and Y. Zhou, Almost $p$-ary perfect sequences,, in Sequences and their Applications - SETA 2010, (2010), 399.  doi: 10.1007/978-3-642-15874-2_34.  Google Scholar [4] T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory, ().   Google Scholar [5] D. Jungnickel and A. Pott, Perfect and almost perfect sequences,, Discrete Appl. Math., 95 (1999), 331.  doi: 10.1016/S0166-218X(99)00085-2.  Google Scholar [6] S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences,, Int. J. Inf. Coding Theory, 1 (2009), 15.  doi: 10.1504/IJICOT.2009.024045.  Google Scholar [7] S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices,, J. Algebra, 175 (1995), 505.  doi: 10.1006/jabr.1995.1198.  Google Scholar [8] S. L. Ma and B. Schmidt, On $(p^a,p,p^a,p^{a-1})$-relative difference sets,, Des. Codes Crypt., 6 (1995), 57.  doi: 10.1007/BF01390771.  Google Scholar [9] F. Özbudak, O. Yayla and C. C. Yíldírím, Nonexistence of certain almost $p$-ary perfect sequences,, in Sequences and their Applications - SETA 2012, (2012), 13.  doi: 10.1007/978-3-642-30615-0_2.  Google Scholar [10] A. Pott, Finite Geometry and Character Theory,, Springer-Verlag, (1995).   Google Scholar [11] R. J. Turyn, Character sums and difference sets,, Pacific J. Math., 15 (1965), 319.   Google Scholar [12] A. Winterhof, O. Yayla and V. Ziegler, Non-existence of some nearly perfect sequences, near Butson-Hadamard matrices, and near conference matrices,, preprint, ().   Google Scholar

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