Linear spans of optimal sets of frequency hopping sequences
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 3, pp. 343-354.

Frequency hopping sequences sets are required in frequency hopping code division multiple access systems. For the anti-jamming purpose, frequency hopping sequences are required to have a large linear span. In this paper, by using a permutation polynomial δ(x) over a finite field, we transform several optimal sets of frequency hopping sequences with small linear span into ones with large linear span. The exact values of the linear span are presented by using the methods of counting the terms of the sequences representations. The results show that the transformed frequency hopping sequences are optimal with respect to the Peng-Fan bound, and can resist the analysis of Berlekamp-Massey algorithm.

DOI : 10.1051/ita/2012007
Classification : 94A05, 94A55, 94A60
Mots-clés : frequency hopping sequences, linear span, permutation polynomials, optimal sets 
@article{ITA_2012__46_3_343_0,
     author = {Juntao, Gao and Yupu, Hu and Xuelian, Li},
     title = {Linear spans of optimal sets of frequency hopping sequences},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {343--354},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {3},
     year = {2012},
     doi = {10.1051/ita/2012007},
     mrnumber = {2981674},
     zbl = {1256.94007},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2012007/}
}
TY  - JOUR
AU  - Juntao, Gao
AU  - Yupu, Hu
AU  - Xuelian, Li
TI  - Linear spans of optimal sets of frequency hopping sequences
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2012
SP  - 343
EP  - 354
VL  - 46
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2012007/
DO  - 10.1051/ita/2012007
LA  - en
ID  - ITA_2012__46_3_343_0
ER  - 
%0 Journal Article
%A Juntao, Gao
%A Yupu, Hu
%A Xuelian, Li
%T Linear spans of optimal sets of frequency hopping sequences
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2012
%P 343-354
%V 46
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita/2012007/
%R 10.1051/ita/2012007
%G en
%F ITA_2012__46_3_343_0
Juntao, Gao; Yupu, Hu; Xuelian, Li. Linear spans of optimal sets of frequency hopping sequences. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 3, pp. 343-354. doi : 10.1051/ita/2012007. http://www.numdam.org/articles/10.1051/ita/2012007/

[1] M. Antweiler and L. Bömer, Complex sequences over GF(pM) with a two-level autocorrelation function and a large linear span. IEEE Trans. Inf. Theory 38 (1992) 120-30. | MR | Zbl

[2] W. Chu and C.J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory 51 (2005) 1139-1141. | MR | Zbl

[3] C. Ding and J. Yin, Sets of optimal frequency hopping sequences, IEEE Trans. Inf. Theory 54 (2008) 3741-3745. | MR

[4] C. Ding, M. Miosio and J. Yuan, Algebraic constructions of optimal frequency hopping sequences. IEEE Trans. Inf. Theory 53 (2007) 2606-2610. | MR | Zbl

[5] C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences : bounds and optimal constructions. IEEE Trans. Inf. Theory 55 (2009) 3297-3304. | MR

[6] C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 56 (2010) 3605-3612. | MR

[7] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences : a combinatorial approach. IEEE Trans. Inf. Theory 50 (2004) 2408-2420. | MR | Zbl

[8] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency hopping sequences. J. Comb. Th. (A) 113 (2006) 1699-1718. | MR | Zbl

[9] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences : auto- and cross-correlation properties. IEEE Trans. Inf. Theory 55 (2009) 867-879. | MR

[10] S.W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communication, Cryptography, and Radar. Cambridge University, Cambridge, UK Press (2005). | MR | Zbl

[11] J.J. Komo and S.C. Liu, Maximal length sequences for frequency hopping. IEEE J. Select. Areas Commun. 5 (1990) 819-822.

[12] P.V. Kumar, Frequency-hopping code sequence designs having large linear span. IEEE Trans. Inf. Theory 34 (1988) 146-151. | MR

[13] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20 (1974) 90-94. | MR | Zbl

[14] R. Lidl and H. Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, UK 20 (1997). | MR | Zbl

[15] D. Peng and P. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences. IEEE Trans. Inf. Theory 50 (2004) 2149-2154. | MR

[16] M.K. Simon, J.K. Omura, R.A. Scholz and B.K. Levitt, Spread Spectrum communications Handbook. McGraw-Hill, New York (2002).

[17] P. Udaya and M.N. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings. IEEE Trans Inf. Theory 44 (1998) 1492-1503. | MR | Zbl

[18] Q. Wang, Optimal sets of frequency hopping sequences with large linear spans. IEEE Trans. Inf. Theory 56 (2010) 1729-1736. | MR

[19] Z. Zhou and X. Tang, A new construction of optimal frequency hopping sequence sets. IEEE Proc. of IWSDA'09 (2009) 92-95.

Cité par Sources :