New nearly optimal codebooks from relative difference sets

Previous Article
Optimal batch codes: Many items or low retrieval requirement
AMC Home
This Issue
Next Article
On optimal ternary linear codes of dimension 6

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

New nearly optimal codebooks from relative difference sets

Zhengchun Zhou ^1, and Xiaohu Tang ^2,

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
Related Papers

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), 316-331. 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.

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), 316-331. 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.

[1]	Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4259-4278. doi: 10.3934/dcds.2018186
[2]	Büşra Özden, Oǧuz Yayla. Partial direct product difference sets and almost quaternary sequences. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021010
[3]	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
[4]	Joško Mandić, Tanja Vučičić. On the existence of Hadamard difference sets in groups of order 400. Advances in Mathematics of Communications, 2016, 10 (3) : 547-554. doi: 10.3934/amc.2016025
[5]	Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041
[6]	Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175
[7]	Dirk Frettlöh, Christoph Richard. Dynamical properties of almost repetitive Delone sets. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 531-556. doi: 10.3934/dcds.2014.34.531
[8]	Nguyen Van Thoai. Decomposition branch and bound algorithm for optimization problems over efficient sets. Journal of Industrial and Management Optimization, 2008, 4 (4) : 647-660. doi: 10.3934/jimo.2008.4.647
[9]	Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55
[10]	Valentina Taddei. Bound sets for floquet boundary value problems: The nonsmooth case. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 459-473. doi: 10.3934/dcds.2000.6.459
[11]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[12]	Gary Froyland, Philip K. Pollett, Robyn M. Stuart. A closing scheme for finding almost-invariant sets in open dynamical systems. Journal of Computational Dynamics, 2014, 1 (1) : 135-162. doi: 10.3934/jcd.2014.1.135
[13]	Eric Baer, Alessio Figalli. Characterization of isoperimetric sets inside almost-convex cones. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 1-14. doi: 10.3934/dcds.2017001
[14]	Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
[15]	S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111.
[16]	Frank D. Grosshans, Jürgen Scheurle, Sebastian Walcher. Invariant sets forced by symmetry. Journal of Geometric Mechanics, 2012, 4 (3) : 271-296. doi: 10.3934/jgm.2012.4.271
[17]	Piotr Oprocha. Coherent lists and chaotic sets. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 797-825. doi: 10.3934/dcds.2011.31.797
[18]	Arya Mazumdar, Ron M. Roth, Pascal O. Vontobel. On linear balancing sets. Advances in Mathematics of Communications, 2010, 4 (3) : 345-361. doi: 10.3934/amc.2010.4.345
[19]	Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403
[20]	L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57.

2021 Impact Factor: 1.015

Tools

Metrics

Other articles
by authors

on AIMS
Zhengchun Zhou Xiaohu Tang
on Google Scholar
Zhengchun Zhou Xiaohu Tang

[Back to Top]