Nearly perfect sequences with arbitrary out-of-phase autocorrelation

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

Nearly perfect sequences with arbitrary out-of-phase autocorrelation

Oǧuz Yayla ^1,

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

Received September 2014 Published April 2016

Abstract
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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.

Keywords: relative difference set, direct product difference set, Perfect sequence, multiplier., nearly perfect sequence.

Mathematics Subject Classification: Primary: 05B10; Secondary: 94A5.

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, Cambridge Univ. Press, 1999.
[2]	B. W. Brock, Hermitian congruence and the existence and completion of generalized Hadamard matrices, J. Combin. Theory Ser. A, 49 (1988), 233-261. doi: 10.1016/0097-3165(88)90054-4.
[3]	Y. M. Chee, Y. Tan and Y. Zhou, Almost $p$-ary perfect sequences, in Sequences and their Applications - SETA 2010, Springer, Berlin, 2010, 399-415. doi: 10.1007/978-3-642-15874-2_34.
[4]	T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory, ().
[5]	D. Jungnickel and A. Pott, Perfect and almost perfect sequences, Discrete Appl. Math., 95 (1999), 331-359. doi: 10.1016/S0166-218X(99)00085-2.
[6]	S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences, Int. J. Inf. Coding Theory, 1 (2009), 15-38. doi: 10.1504/IJICOT.2009.024045.
[7]	S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices, J. Algebra, 175 (1995), 505-525. doi: 10.1006/jabr.1995.1198.
[8]	S. L. Ma and B. Schmidt, On $(p^a,p,p^a,p^{a-1})$-relative difference sets, Des. Codes Crypt., 6 (1995), 57-71. doi: 10.1007/BF01390771.
[9]	F. Özbudak, O. Yayla and C. C. Yíldírím, Nonexistence of certain almost $p$-ary perfect sequences, in Sequences and their Applications - SETA 2012, Springer, Heidelberg, 2012, 13-24. doi: 10.1007/978-3-642-30615-0_2.
[10]	A. Pott, Finite Geometry and Character Theory, Springer-Verlag, Berlin, 1995.
[11]	R. J. Turyn, Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.
[12]	A. Winterhof, O. Yayla and V. Ziegler, Non-existence of some nearly perfect sequences, near Butson-Hadamard matrices, and near conference matrices,, preprint, ().

show all references

References:

[1]	T. Beth, D. Jungnickel and H. Lenz, Design Theory, 2nd edition, Cambridge Univ. Press, 1999.
[2]	B. W. Brock, Hermitian congruence and the existence and completion of generalized Hadamard matrices, J. Combin. Theory Ser. A, 49 (1988), 233-261. doi: 10.1016/0097-3165(88)90054-4.
[3]	Y. M. Chee, Y. Tan and Y. Zhou, Almost $p$-ary perfect sequences, in Sequences and their Applications - SETA 2010, Springer, Berlin, 2010, 399-415. doi: 10.1007/978-3-642-15874-2_34.
[4]	T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory, ().
[5]	D. Jungnickel and A. Pott, Perfect and almost perfect sequences, Discrete Appl. Math., 95 (1999), 331-359. doi: 10.1016/S0166-218X(99)00085-2.
[6]	S. L. Ma and W. S. Ng, On non-existence of perfect and nearly perfect sequences, Int. J. Inf. Coding Theory, 1 (2009), 15-38. doi: 10.1504/IJICOT.2009.024045.
[7]	S. L. Ma and A. Pott, Relative difference sets, planar functions, and generalized Hadamard matrices, J. Algebra, 175 (1995), 505-525. doi: 10.1006/jabr.1995.1198.
[8]	S. L. Ma and B. Schmidt, On $(p^a,p,p^a,p^{a-1})$-relative difference sets, Des. Codes Crypt., 6 (1995), 57-71. doi: 10.1007/BF01390771.
[9]	F. Özbudak, O. Yayla and C. C. Yíldírím, Nonexistence of certain almost $p$-ary perfect sequences, in Sequences and their Applications - SETA 2012, Springer, Heidelberg, 2012, 13-24. doi: 10.1007/978-3-642-30615-0_2.
[10]	A. Pott, Finite Geometry and Character Theory, Springer-Verlag, Berlin, 1995.
[11]	R. J. Turyn, Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.
[12]	A. Winterhof, O. Yayla and V. Ziegler, Non-existence of some nearly perfect sequences, near Butson-Hadamard matrices, and near conference matrices,, preprint, ().

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