# American Institute of Mathematical Sciences

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

 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.

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. [2] S. Bozta¸s and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010,2010,1300-1304. [3] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996. [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. [5] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411. doi: 10.1109/18.370141. [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. [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. [8] E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004,2005,197-207. doi: 10.1007/11423461_13. [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. [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. [11] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498. [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. [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. [14] A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048. [15] X. H. Tang, A note on d-form function with difference balanced property, preprint. [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. [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.

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. [2] S. Bozta¸s and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010,2010,1300-1304. [3] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, London, 1996. [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. [5] G. Gong, Theory and applications of q-ary interleaved sequences, IEEE Trans. Inf. Theory, 41 (1995), 400-411. doi: 10.1109/18.370141. [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. [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. [8] E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004,2005,197-207. doi: 10.1007/11423461_13. [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. [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. [11] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498. [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. [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. [14] A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048. [15] X. H. Tang, A note on d-form function with difference balanced property, preprint. [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. [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.
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