# American Institute of Mathematical Sciences

February  2014, 8(1): 83-101. doi: 10.3934/amc.2014.8.83

## Sets of zero-difference balanced functions and their applications

 1 Institute of Algebra and Geometry, Otto-von-Guericke University Magdeburg, 39106 Magdeburg, Germany 2 Department of Mathematics and System Sciences, National University of Defense Technology, Changsha, Hunan 410073, China

Received  March 2013 Revised  September 2013 Published  January 2014

Zero-difference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.
Citation: Qi Wang, Yue Zhou. Sets of zero-difference balanced functions and their applications. Advances in Mathematics of Communications, 2014, 8 (1) : 83-101. doi: 10.3934/amc.2014.8.83
##### References:
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Kumar, Frequency-hopping code sequence designs having large linear span,, IEEE Trans. Inform. Theory, 34 (1988), 146.  doi: 10.1109/18.2616.  Google Scholar [18] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90.   Google Scholar [19] V. I. Levenšteĭn, A certain method of constructing quasilinear codes that guarantee synchronization in the presence of errors,, Problemy Peredači Informacii, 7 (1971), 30.   Google Scholar [20] V. I. Levenšteĭn, Combinatorial problems motivated by comma-free codes,, J. Combin. Des., 12 (2004), 184.  doi: 10.1002/jcd.10071.  Google Scholar [21] R. Lidl and H. Niederreiter, Finite Fields, Second edition,, Cambridge University Press, (1997).   Google Scholar [22] Y. Luo, F.-W. Fu, A. J. H. Vinck and W. Chen, On constant-composition codes over $Z_q$,, IEEE Trans. Inform. Theory, 49 (2003), 3010.  doi: 10.1109/TIT.2003.819339.  Google Scholar [23] H. Niederreiter and A. Winterhof, Cyclotomic $\mathfrakR$-orthomorphisms of finite fields,, Discrete Math., 295 (2005), 161.  doi: 10.1016/j.disc.2004.12.011.  Google Scholar [24] K. Nyberg, Perfect nonlinear S-boxes,, in Advances in Cryptology-EUROCRYPT '91, (1991), 378.  doi: 10.1007/3-540-46416-6_32.  Google Scholar [25] D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2149.  doi: 10.1109/TIT.2004.833362.  Google Scholar [26] D. V. Sarwate, Comments on "Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences'' by D. Peng and P. Fan,, IEEE Trans. Inform. Theory, 51 (2005).  doi: 10.1109/TIT.2005.844055.  Google Scholar [27] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, revised edition,, McGraw-Hill Inc., (2002).   Google Scholar [28] H. Wang, A new bound for difference systems of sets,, J. Combin. Math. Combin. Comput., 58 (2006), 161.   Google Scholar [29] Q. Wang, Optimal sets of frequency hopping sequences with large linear spans,, IEEE Trans. Inform. Theory, 56 (2010), 1729.  doi: 10.1109/TIT.2010.2040874.  Google Scholar [30] Z. Zhou, X. Tang, D. Wu and Y. Yang, Some new classes of zero-difference balanced functions,, IEEE Trans. Inform. Theory, 58 (2012), 139.  doi: 10.1109/TIT.2011.2171418.  Google Scholar

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##### References:
 [1] M. Antweiler and L. Bömer, Complex sequences over GF$(p^M)$ with a two-level autocorrelation function and a large linear span,, IEEE Trans. Inform. Theory, (1992), 120.  doi: 10.1109/18.108256.  Google Scholar [2] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes,, IEEE Trans. Inform. Theory, (2005), 2089.  doi: 10.1109/TIT.2005.847722.  Google Scholar [3] L. Carlitz and C. Wells, The number of solutions of a special system of equations in a finite field,, Acta Arith., 12 (): 77.   Google Scholar [4] P. Dembowski and T. G. Ostrom, Planes of order $n$ with collineation groups of order $n^2$,, Math. Z., 103 (1968), 239.  doi: 10.1007/BF01111042.  Google Scholar [5] C. Ding, Optimal constant composition codes from zero-difference balanced functions,, IEEE Trans. Inform. Theory, 54 (2008), 5766.  doi: 10.1109/TIT.2008.2006420.  Google Scholar [6] C. Ding, Optimal and perfect difference systems of sets,, J. Combin. Theory Ser. A, 116 (2009), 109.  doi: 10.1016/j.jcta.2008.05.007.  Google Scholar [7] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 53 (2007), 2606.  doi: 10.1109/TIT.2007.899545.  Google Scholar [8] C. Ding and Y. Tan, Zero-difference balanced functions with applications,, J. Stat. Theory Practice, 6 (2012), 3.  doi: 10.1080/15598608.2012.647479.  Google Scholar [9] C. Ding and J. Yin, Algebraic constructions of constant composition codes,, IEEE Trans. Inform. Theory, 51 (2005), 1585.  doi: 10.1109/TIT.2005.844087.  Google Scholar [10] C. Ding and J. Yin, Combinatorial constructions of optimal constant-composition codes,, IEEE Trans. Inform. Theory, 51 (2005), 3671.  doi: 10.1109/TIT.2005.855612.  Google Scholar [11] C. Ding and J. Yin, Sets of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 54 (2008), 3741.  doi: 10.1109/TIT.2008.926410.  Google Scholar [12] F.-W. Fu, A. J. H. Vinck and S.-Y. Shen, On the constructions of constant-weight codes,, IEEE Trans. Inform. Theory, 44 (1998), 328.  doi: 10.1109/18.651060.  Google Scholar [13] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach,, IEEE Trans. Inform. Theory, 50 (2004), 2408.  doi: 10.1109/TIT.2004.834783.  Google Scholar [14] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency-hopping sequences,, J. Combin. Theory Ser. A, 113 (2006), 1699.  doi: 10.1016/j.jcta.2006.03.019.  Google Scholar [15] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: auto- and cross-correlation properties,, IEEE Trans. Inform. Theory, 55 (2009), 867.  doi: 10.1109/TIT.2008.2009856.  Google Scholar [16] S. W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communication, Cryptography, and Radar,, Cambridge University Press, (2005).  doi: 10.1017/CBO9780511546907.  Google Scholar [17] P. V. Kumar, Frequency-hopping code sequence designs having large linear span,, IEEE Trans. Inform. Theory, 34 (1988), 146.  doi: 10.1109/18.2616.  Google Scholar [18] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90.   Google Scholar [19] V. I. Levenšteĭn, A certain method of constructing quasilinear codes that guarantee synchronization in the presence of errors,, Problemy Peredači Informacii, 7 (1971), 30.   Google Scholar [20] V. I. Levenšteĭn, Combinatorial problems motivated by comma-free codes,, J. Combin. Des., 12 (2004), 184.  doi: 10.1002/jcd.10071.  Google Scholar [21] R. Lidl and H. Niederreiter, Finite Fields, Second edition,, Cambridge University Press, (1997).   Google Scholar [22] Y. Luo, F.-W. Fu, A. J. H. Vinck and W. Chen, On constant-composition codes over $Z_q$,, IEEE Trans. Inform. Theory, 49 (2003), 3010.  doi: 10.1109/TIT.2003.819339.  Google Scholar [23] H. Niederreiter and A. Winterhof, Cyclotomic $\mathfrakR$-orthomorphisms of finite fields,, Discrete Math., 295 (2005), 161.  doi: 10.1016/j.disc.2004.12.011.  Google Scholar [24] K. Nyberg, Perfect nonlinear S-boxes,, in Advances in Cryptology-EUROCRYPT '91, (1991), 378.  doi: 10.1007/3-540-46416-6_32.  Google Scholar [25] D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences,, IEEE Trans. Inform. Theory, 50 (2004), 2149.  doi: 10.1109/TIT.2004.833362.  Google Scholar [26] D. V. Sarwate, Comments on "Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences'' by D. Peng and P. Fan,, IEEE Trans. Inform. Theory, 51 (2005).  doi: 10.1109/TIT.2005.844055.  Google Scholar [27] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, revised edition,, McGraw-Hill Inc., (2002).   Google Scholar [28] H. Wang, A new bound for difference systems of sets,, J. Combin. Math. Combin. Comput., 58 (2006), 161.   Google Scholar [29] Q. Wang, Optimal sets of frequency hopping sequences with large linear spans,, IEEE Trans. Inform. Theory, 56 (2010), 1729.  doi: 10.1109/TIT.2010.2040874.  Google Scholar [30] Z. Zhou, X. Tang, D. Wu and Y. Yang, Some new classes of zero-difference balanced functions,, IEEE Trans. Inform. Theory, 58 (2012), 139.  doi: 10.1109/TIT.2011.2171418.  Google Scholar
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