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February  2017, 11(1): 67-76. doi: 10.3934/amc.2017002

New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property

Yang Yang 1,2, , Xiaohu Tang 3, and  Guang Gong 4,

1. 

Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
2. 

Science and Technology on Communication Security Laboratory, Maibox 810, Chengdu, Sichuan 610041, China
3. 

Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
4. 

Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received  April 2015 Published  February 2017

Fund Project: The work of Y. Yang and X.H. Tang was supported in part by the National Science Foundation of China (NSFC) under Grants 61401376 and 61171095, in part by Science and Technology on Communication Security Laboratory Grant 9140C110302150C11004. This paper was presented in part at the 7th International Conference on Sequences and Their Applications [16], SETA 2012, Waterloo, Canada, June 4-8, 2012.

By using shift sequences defined by difference balanced functions with d-form property, and column sequences defined by a mutually orthogonal almost perfect sequences pair, new almost perfect, odd perfect, and perfect sequences are obtained via interleaving method. Furthermore, the proposed perfect QAM+ sequences positively answer to the problem of the existence of perfect QAM+ sequences proposed by Boztaş and Udaya.

Keywords: Almost perfect sequences, perfect sequences, odd perfect sequences, QAM+ sequences, difference balanced function, d-form..
Mathematics Subject Classification: Primary: 94A05, 60G35.
Citation: 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
References:
[1]

M. Antweiler, Cross-correlation of p-ary GMW sequences, IEEE Trans. Inf. Theory, 40 (1994), 1253-1261.  doi: 10.1109/18.335941.  Google Scholar

[2]

S. Bozta¸s and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010,2010,1300-1304. Google Scholar

[3] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996.   Google Scholar
[4] S. W. Golomb and G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar
[5]

G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411.  doi: 10.1109/18.370141.  Google Scholar

[6]

T. Helleseth and G. Gong, New binary sequences with ideal-level autocorrelation function, IEEE Trans. Inf. Theory, 154 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[7]

A. Klapper, d-form sequence: Families of sequences with low correlaltion values and large linear spans, IEEE Trans. Inf. Theory, 51 (1995), 1469-1477.  doi: 10.1109/18.370143.  Google Scholar

[8]

E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004,2005,197-207. doi: 10.1007/11423461_13.  Google Scholar

[9]

C. E. Lee, On a New Class of 5-ary Sequences Exhibiting Ideal Periodic Autocorrelation Properties with Applications to Spread Specturm Systems, Ph. D thesis, Mississipi State Univ. , 1986. Google Scholar

[10]

C. E. Lee, Perfect q-ary sequences from multiplicative characters over GF (p), Electr. lett., 28 (1992), 833-834.  doi: 10.1049/el:19920527.  Google Scholar

[11]

H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498.   Google Scholar

[12]

W. H. Mow, Even-odd transormation with application to multi-user CW radars, in 1996 IEEE 4th Int. Symp. Spread Spectrum Techn. Appl. Proc. , Mainz, 1996,191-193. Google Scholar

[13]

J.-S. No, New cyclic diffrence sets with Singer parameters constructed from d-homogeneous function, Des. Codes Cryptogr., 33 (2004), 199-213.  doi: 10.1023/B:DESI.0000036246.52472.81.  Google Scholar

[14]

A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861.  doi: 10.1016/j.disc.2006.06.048.  Google Scholar

[15]

X. H. Tang, A note on d-form function with difference balanced property, preprint. Google Scholar

[16]

Y. Yang, G. Gong and X. H. Tang, Odd perfect sequences and sets of spreading sequences with zero or low odd periodic correlation zone, in SETA 2012,2012, 1-12. doi: 10.1007/978-3-642-30615-0_1.  Google Scholar

[17]

X. Y. Zeng, L. Hu and Q. C. Liu, A novel method for constructing almost perfect polyphase sequences, in WCC 2005,2006,346-353. doi: 10.1007/11779360_27.  Google Scholar

show all references

References:
[1]

M. Antweiler, Cross-correlation of p-ary GMW sequences, IEEE Trans. Inf. Theory, 40 (1994), 1253-1261.  doi: 10.1109/18.335941.  Google Scholar

[2]

S. Bozta¸s and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010,2010,1300-1304. Google Scholar

[3] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996.   Google Scholar
[4] S. W. Golomb and G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.  Google Scholar
[5]

G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411.  doi: 10.1109/18.370141.  Google Scholar

[6]

T. Helleseth and G. Gong, New binary sequences with ideal-level autocorrelation function, IEEE Trans. Inf. Theory, 154 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[7]

A. Klapper, d-form sequence: Families of sequences with low correlaltion values and large linear spans, IEEE Trans. Inf. Theory, 51 (1995), 1469-1477.  doi: 10.1109/18.370143.  Google Scholar

[8]

E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004,2005,197-207. doi: 10.1007/11423461_13.  Google Scholar

[9]

C. E. Lee, On a New Class of 5-ary Sequences Exhibiting Ideal Periodic Autocorrelation Properties with Applications to Spread Specturm Systems, Ph. D thesis, Mississipi State Univ. , 1986. Google Scholar

[10]

C. E. Lee, Perfect q-ary sequences from multiplicative characters over GF (p), Electr. lett., 28 (1992), 833-834.  doi: 10.1049/el:19920527.  Google Scholar

[11]

H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498.   Google Scholar

[12]

W. H. Mow, Even-odd transormation with application to multi-user CW radars, in 1996 IEEE 4th Int. Symp. Spread Spectrum Techn. Appl. Proc. , Mainz, 1996,191-193. Google Scholar

[13]

J.-S. No, New cyclic diffrence sets with Singer parameters constructed from d-homogeneous function, Des. Codes Cryptogr., 33 (2004), 199-213.  doi: 10.1023/B:DESI.0000036246.52472.81.  Google Scholar

[14]

A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861.  doi: 10.1016/j.disc.2006.06.048.  Google Scholar

[15]

X. H. Tang, A note on d-form function with difference balanced property, preprint. Google Scholar

[16]

Y. Yang, G. Gong and X. H. Tang, Odd perfect sequences and sets of spreading sequences with zero or low odd periodic correlation zone, in SETA 2012,2012, 1-12. doi: 10.1007/978-3-642-30615-0_1.  Google Scholar

[17]

X. Y. Zeng, L. Hu and Q. C. Liu, A novel method for constructing almost perfect polyphase sequences, in WCC 2005,2006,346-353. doi: 10.1007/11779360_27.  Google Scholar

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