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.  doi: 10.1016/0097-3165(95)90006-3.  Google Scholar

[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.  Google Scholar

[3]

R. C. Bose, An affine analog of Singer's theorem,, J. Indian Math. Soc., 6 (1942), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

C. Ding, Complex codebooks from combinatorial designs,, IEEE Trans. Inform. Theory, 52 (2006), 4229.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[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.  Google Scholar

[8]

C. Ding and T. Feng, Codebooks from almost difference sets,, Des. Codes Cryptogr., 46 (2008), 113.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Lidl and H. Niederreiter, "Finite Fields,'', Addison-Wesley, (1983).   Google Scholar

[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.  Google Scholar

[14]

J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems,, in, (1993), 63.   Google Scholar

[15]

A. Pott, "Finite Geometry and Character Theory,'', Springer, (1995).   Google Scholar

[16]

A. Pott, A survey on relative difference sets,, in, (1996), 195.   Google Scholar

[17]

D. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

R. C. Bose, An affine analog of Singer's theorem,, J. Indian Math. Soc., 6 (1942), 1.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

C. Ding, Complex codebooks from combinatorial designs,, IEEE Trans. Inform. Theory, 52 (2006), 4229.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[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.  Google Scholar

[8]

C. Ding and T. Feng, Codebooks from almost difference sets,, Des. Codes Cryptogr., 46 (2008), 113.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Lidl and H. Niederreiter, "Finite Fields,'', Addison-Wesley, (1983).   Google Scholar

[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.  Google Scholar

[14]

J. L. Massey and T. Mittlelholzer, Welch's bound and sequence sets for code-division multiple-access systems,, in, (1993), 63.   Google Scholar

[15]

A. Pott, "Finite Geometry and Character Theory,'', Springer, (1995).   Google Scholar

[16]

A. Pott, A survey on relative difference sets,, in, (1996), 195.   Google Scholar

[17]

D. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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