American Institute of Mathematical Sciences

August  2011, 5(3): 521-527. doi: 10.3934/amc.2011.5.521

New nearly optimal codebooks from relative difference sets

 1 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China 2 Provincial Key Lab of Information Coding and Transmission, Southwest Jiaotong University, Chengdu, 610031, China

Received  December 2010 Published  August 2011

Codebooks achieving the Welch bound on the maximum correlation amplitude are desirable in a number of applications. Recently, codebooks meeting (resp., nearly meeting) the Welch bound were constructed from difference sets (resp., almost difference sets). In this paper, a general connection between complex codebooks and relative difference sets is introduced. Several classes of codebooks nearly meeting the Welch bound are then constructed from some known relative difference sets using the general connection.
Citation: 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
References:
 [1] K. T. Arasu, J. F. Dillon, D. Jungnickel and A. Pott, The solution of the Waterloo problem, J. Combin. Theory Ser. A, 71 (1995), 316-331. doi: 10.1016/0097-3165(95)90006-3. [2] K. T. Arasu, J. F. Dillon, K. H. Leung and S. L. Ma, Cyclic relative difference sets with classical parameters, J. Combin. Theory Ser. A, 94 (2001), 118-126. doi: 10.1006/jcta.2000.3137. [3] R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., 6 (1942), 1-15. [4] D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks, Des. Codes Cryptogr., 30 (2003), 325-343. doi: 10.1023/A:1025750228679. [5] J. H. Conway, R. H. Harding and N. J. A. Sloane, Packing lines, planes, etc.: Packings in grassmannian spaces, Exp. Math., 5 (1996), 139-159. [6] C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inform. Theory, 52 (2006), 4229-4235. doi: 10.1109/TIT.2006.880058. [7] C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inform. Theory, 53 (2007), 4245-4250. doi: 10.1109/TIT.2007.907343. [8] C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126. doi: 10.1007/s10623-007-9140-z. [9] S.-H. Kim, J.-S. No, H.-C. Chung and T. Helleseth, New cyclic relative difference sets constructed from $d$-homogeneous functions with difference-balanced property, IEEE Trans. Inform. Theory, 51 (2005), 1155-1163. [10] P. V. Kumar, On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function, IEEE Trans. Inform. Theory, 34 (1988), 271-277. doi: 10.1109/18.2635. [11] K. H. Leung and S. L. Ma, Constructions of relative difference sets with classical parameters and circulant weighing matrices, J. Combin. Theory Ser. A, 99 (2002), 111-127 . doi: 10.1006/jcta.2002.3262. [12] R. Lidl and H. Niederreiter, "Finite Fields,'' Addison-Wesley, Reading, MA, 1983. [13] 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. [14] J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems, in "Sequences II: Methods in Communication, Security and Computer Science,'' Springer, Heidelberg, New York, (1993), 63-78. [15] A. Pott, "Finite Geometry and Character Theory,'' Springer, 1995. [16] A. Pott, A survey on relative difference sets, in "Groups, Difference Sets, and the Monster'' (eds. K.T. Arasu, et al.), Walter de Gruyter, (1996), 195-232. [17] D. Sarwate, Meeting the Welch bound with equality, in "Proc. of SETA'98,'' Springer-Verlag, Berlin, Heidelberg, (1999), 79-102. [18] T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmonic Anal., 14 (2003), 257-275. doi: 10.1016/S1063-5203(03)00023-X. [19] L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, IT-20 (1974), 397-399. doi: 10.1109/TIT.1974.1055219. [20] P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory, 51 (2005), 1900-1907. doi: 10.1109/TIT.1974.1055219.

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References:
 [1] K. T. Arasu, J. F. Dillon, D. Jungnickel and A. Pott, The solution of the Waterloo problem, J. Combin. Theory Ser. A, 71 (1995), 316-331. doi: 10.1016/0097-3165(95)90006-3. [2] K. T. Arasu, J. F. Dillon, K. H. Leung and S. L. Ma, Cyclic relative difference sets with classical parameters, J. Combin. Theory Ser. A, 94 (2001), 118-126. doi: 10.1006/jcta.2000.3137. [3] R. C. Bose, An affine analog of Singer's theorem, J. Indian Math. Soc., 6 (1942), 1-15. [4] D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks, Des. Codes Cryptogr., 30 (2003), 325-343. doi: 10.1023/A:1025750228679. [5] J. H. Conway, R. H. Harding and N. J. A. Sloane, Packing lines, planes, etc.: Packings in grassmannian spaces, Exp. Math., 5 (1996), 139-159. [6] C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inform. Theory, 52 (2006), 4229-4235. doi: 10.1109/TIT.2006.880058. [7] C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inform. Theory, 53 (2007), 4245-4250. doi: 10.1109/TIT.2007.907343. [8] C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126. doi: 10.1007/s10623-007-9140-z. [9] S.-H. Kim, J.-S. No, H.-C. Chung and T. Helleseth, New cyclic relative difference sets constructed from $d$-homogeneous functions with difference-balanced property, IEEE Trans. Inform. Theory, 51 (2005), 1155-1163. [10] P. V. Kumar, On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function, IEEE Trans. Inform. Theory, 34 (1988), 271-277. doi: 10.1109/18.2635. [11] K. H. Leung and S. L. Ma, Constructions of relative difference sets with classical parameters and circulant weighing matrices, J. Combin. Theory Ser. A, 99 (2002), 111-127 . doi: 10.1006/jcta.2002.3262. [12] R. Lidl and H. Niederreiter, "Finite Fields,'' Addison-Wesley, Reading, MA, 1983. [13] 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. [14] J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems, in "Sequences II: Methods in Communication, Security and Computer Science,'' Springer, Heidelberg, New York, (1993), 63-78. [15] A. Pott, "Finite Geometry and Character Theory,'' Springer, 1995. [16] A. Pott, A survey on relative difference sets, in "Groups, Difference Sets, and the Monster'' (eds. K.T. Arasu, et al.), Walter de Gruyter, (1996), 195-232. [17] D. Sarwate, Meeting the Welch bound with equality, in "Proc. of SETA'98,'' Springer-Verlag, Berlin, Heidelberg, (1999), 79-102. [18] T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication, Appl. Comput. Harmonic Anal., 14 (2003), 257-275. doi: 10.1016/S1063-5203(03)00023-X. [19] L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Trans. Inform. Theory, IT-20 (1974), 397-399. doi: 10.1109/TIT.1974.1055219. [20] P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inform. Theory, 51 (2005), 1900-1907. doi: 10.1109/TIT.1974.1055219.
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