New nearly optimal codebooks from relative difference sets

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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

Abstract
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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.

Keywords: Welch bound, Codebooks, difference sets, relative dierence sets., almost difference sets, signal sets.

Mathematics Subject Classification: Primary: 94A05, 94B60; Secondary: 05B1.

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. 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. doi: 10.1006/jcta.2000.3137.
[3]	R. C. Bose, An affine analog of Singer's theorem,, J. Indian Math. Soc., 6 (1942), 1.
[4]	D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks,, Des. Codes Cryptogr., 30 (2003), 325. 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.
[6]	C. Ding, Complex codebooks from combinatorial designs,, IEEE Trans. Inform. Theory, 52 (2006), 4229. 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. doi: 10.1109/TIT.2007.907343.
[8]	C. Ding and T. Feng, Codebooks from almost difference sets,, Des. Codes Cryptogr., 46 (2008), 113. 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.
[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. 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. doi: 10.1006/jcta.2002.3262.
[12]	R. Lidl and H. Niederreiter, "Finite Fields,'', Addison-Wesley, (1983).
[13]	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.
[14]	J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems,, in, (1993), 63.
[15]	A. Pott, "Finite Geometry and Character Theory,'', Springer, (1995).
[16]	A. Pott, A survey on relative difference sets,, in, (1996), 195.
[17]	D. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79.
[18]	T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication,, Appl. Comput. Harmonic Anal., 14 (2003), 257. 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. 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. doi: 10.1109/TIT.1974.1055219.

show all references

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. 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. doi: 10.1006/jcta.2000.3137.
[3]	R. C. Bose, An affine analog of Singer's theorem,, J. Indian Math. Soc., 6 (1942), 1.
[4]	D. Chandler and Q. Xiang, Cyclic relative difference sets and their p-ranks,, Des. Codes Cryptogr., 30 (2003), 325. 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.
[6]	C. Ding, Complex codebooks from combinatorial designs,, IEEE Trans. Inform. Theory, 52 (2006), 4229. 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. doi: 10.1109/TIT.2007.907343.
[8]	C. Ding and T. Feng, Codebooks from almost difference sets,, Des. Codes Cryptogr., 46 (2008), 113. 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.
[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. 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. doi: 10.1006/jcta.2002.3262.
[12]	R. Lidl and H. Niederreiter, "Finite Fields,'', Addison-Wesley, (1983).
[13]	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.
[14]	J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems,, in, (1993), 63.
[15]	A. Pott, "Finite Geometry and Character Theory,'', Springer, (1995).
[16]	A. Pott, A survey on relative difference sets,, in, (1996), 195.
[17]	D. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79.
[18]	T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication,, Appl. Comput. Harmonic Anal., 14 (2003), 257. 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. 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. doi: 10.1109/TIT.1974.1055219.

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