Nearly perfect sequences with arbitrary out-of-phase autocorrelation

Previous Article
Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code
AMC Home
This Issue
Next Article
On the error distance of extended Reed-Solomon codes

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

Abstract
Related Papers

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, (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

show all references

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

[1]	Yang Yang, Xiaohu Tang, Guang Gong. New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property. Advances in Mathematics of Communications, 2017, 11 (1) : 67-76. doi: 10.3934/amc.2017002
[2]	Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115
[3]	Zhengchun Zhou, Xiaohu Tang. New nearly optimal codebooks from relative difference sets. Advances in Mathematics of Communications, 2011, 5 (3) : 521-527. doi: 10.3934/amc.2011.5.521
[4]	Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9
[5]	Longye Wang, Gaoyuan Zhang, Hong Wen, Xiaoli Zeng. An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length. Advances in Mathematics of Communications, 2018, 12 (3) : 541-552. doi: 10.3934/amc.2018032
[6]	Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041
[7]	Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[8]	Tomáš Roubíček. Thermodynamics of perfect plasticity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 193-214. doi: 10.3934/dcdss.2013.6.193
[9]	Olof Heden. A survey of perfect codes. Advances in Mathematics of Communications, 2008, 2 (2) : 223-247. doi: 10.3934/amc.2008.2.223
[10]	Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov. Perfect retroreflectors and billiard dynamics. Journal of Modern Dynamics, 2011, 5 (1) : 33-48. doi: 10.3934/jmd.2011.5.33
[11]	Marcela Mejía, J. Urías. An asymptotically perfect pseudorandom generator. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 115-126. doi: 10.3934/dcds.2001.7.115
[12]	Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[13]	Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems & Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[14]	Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi. Hereditarily non uniformly perfect sets. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2391-2402. doi: 10.3934/dcdss.2019150
[15]	Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[16]	Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial & Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451
[17]	Nguyen Thi Bach Kim, Nguyen Canh Nam, Le Quang Thuy. An outcome space algorithm for minimizing the product of two convex functions over a convex set. Journal of Industrial & Management Optimization, 2013, 9 (1) : 243-253. doi: 10.3934/jimo.2013.9.243
[18]	Lan Wen. On the preperiodic set. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[19]	A. V. Borisov, I.S. Mamaev, S. M. Ramodanov. Dynamics of two interacting circular cylinders in perfect fluid. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 235-253. doi: 10.3934/dcds.2007.19.235
[20]	Yves Edel, Alexander Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications, 2009, 3 (1) : 59-81. doi: 10.3934/amc.2009.3.59

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences