Abstract
Let V = {1, 2, . . . , M} and let \({\{H_i{:} i \in V\}}\) be a set of Hadamard matrices with the property that the magnitude of the dot product of any two rows of distinct matrices is bounded above. A Hadamard partition is any partition of the set of all rows of the matrices H i into Hadamard matrices. Such partitions have an application to the security of quasi-synchronous code-division multiple-access radio systems when loosely synchronized (LS) codes are used as spreading codes. A new generation of LS code can be used for each information bit to be spread. For each generation, a Hadamard matrix from some partition is selected for use in the code construction. This code evolution increases security against eavesdropping and jamming. One security aspect requires that the number of Hadamard partitions be large. Thus the number of partitions is studied here. If a Kerdock code construction is used for the set of matrices, the Hadamard partition constructed is shown to be unique. It is also shown here that this is not the case if a Gold (or Gold-like) code construction is used. In this case the number of Hadamard partitions can be enumerated, and is very large.
Similar content being viewed by others
References
Fan P.Z.: Spreading sequence design and theoretical limits for quasisynchronous CDMA systems. EURASIP J. Wireless Commun. Networking 1, 19–31 (2004)
Godsil C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam, The Netherlands (1977).
Sanusi S.O.: Assignment of spreading codes in code division multiple access radio systems. PhD thesis, University of Glamorgan (2006).
Sarwate D.V., Pursley M.B.: Crosscorrelation properties of pseudorandom sequences. Proc. IEEE 68(5), 593–619 (1980)
Stańczak S., Boche H., Haardt M.: Are LAS codes a miracle? In: 2001 IEEE Global Communications Conference (GLOBECOM), BWS05-4, San Antonio, TX, USA (November 25–29), pp. 589–593 (2001).
Tang X., Mow W.H.: Design of spreading codes for quasi-synchronous CDMA with intercell interference. IEEE J. Select. Area Commun. 24(1), 84–93 (2006)
Tseng C.C., Liu C.L.: Complementary sets of sequences. IEEE Trans. Inf. Theory 18(5), 644–652 (1972)
Ward R.P.: Evolution of loosely synchronized spreading codes in code-division multiple-access systems. PhD thesis, University of Glamorgan (2008).
Yang K., Kim Y.-K., Kumar P.V.: Quasi-orthogonal sequences for code-division multiple-access systems. IEEE Trans. Inf. Theory 46(3), 982–993 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Teirlinck.
Rights and permissions
About this article
Cite this article
Smith, D.H., Ward, R.P. & Perkins, S. Gold codes, Hadamard partitions and the security of CDMA systems. Des. Codes Cryptogr. 51, 231–243 (2009). https://doi.org/10.1007/s10623-008-9257-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9257-8