# American Institute of Mathematical Sciences

February  2018, 12(1): 181-188. doi: 10.3934/amc.2018012

## A new family of one-coincidence sets of sequences with dispersed elements for frequency hopping cdma systems

 1 Department of Mathematics, Claremont McKenna College, 850 Columbia Avenue, Claremont, CA 91711, USA 2 Department of Engineering, Harvey Mudd College, Claremont, CA 91711, USA

Received  January 2017 Revised  October 2017 Published  March 2018

We present a new family of one-coincidence sequence sets suitable for frequency hopping code division multiple access (FH-CDMA) systems with dispersed (low density) sequence elements. These sets are derived from one-coincidence prime sequence sets, such that for each one-coincidence prime sequence set there is a new one-coincidence set comprised of sequences with dispersed sequence elements, required in some circumstances, for FH-CDMA systems. Getting rid of crowdedness of sequence elements is achieved by doubling the size of the sequence element alphabet. In addition, this doubling process eases control over the distance between adjacent sequence elements. Properties of the new sets are discussed.

Citation: Lenny Fukshansky, Ahmad A. Shaar. A new family of one-coincidence sets of sequences with dispersed elements for frequency hopping cdma systems. Advances in Mathematics of Communications, 2018, 12 (1) : 181-188. doi: 10.3934/amc.2018012
##### References:
 [1] L. Bin, One-coincidence sequences with specified distance between adjacent symbols for frequency-hopping multiple access, IEEE Trans. Commun., 45 (1997), 408-410.   Google Scholar [2] S. Geirhofer, J. Z. Sun, L. Tong and B. M. Sadler, Cognitive frequency hopping based on interference prediction: Theory and experimental results, ACM SIGMOBILE Mobile Comp. Commun. Review, 13 (2009), 49-61.   Google Scholar [3] C. Hodgdon, Adaptive frequency hopping for reduced interference between bluetooth and wireless LAN, Ericsson Technology Licensing, 2003. Google Scholar [4] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94.   Google Scholar [5] A. A. Shaar and P. A. Davies, A survey of one-coincidence sequences for frequency-hopped spread-spectrum systems, IEEE Proc. F - Commun. Radar Signal Processing, 131 (1984), 719-724.   Google Scholar

show all references

##### References:
 [1] L. Bin, One-coincidence sequences with specified distance between adjacent symbols for frequency-hopping multiple access, IEEE Trans. Commun., 45 (1997), 408-410.   Google Scholar [2] S. Geirhofer, J. Z. Sun, L. Tong and B. M. Sadler, Cognitive frequency hopping based on interference prediction: Theory and experimental results, ACM SIGMOBILE Mobile Comp. Commun. Review, 13 (2009), 49-61.   Google Scholar [3] C. Hodgdon, Adaptive frequency hopping for reduced interference between bluetooth and wireless LAN, Ericsson Technology Licensing, 2003. Google Scholar [4] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94.   Google Scholar [5] A. A. Shaar and P. A. Davies, A survey of one-coincidence sequences for frequency-hopped spread-spectrum systems, IEEE Proc. F - Commun. Radar Signal Processing, 131 (1984), 719-724.   Google Scholar
Set of prime sequences for $p = 7$
 $j \in \mathbb{F}_7$ 0 1 2 3 4 5 6 Sequence $S_0=0 \cdot_7 j$ 0 0 0 0 0 0 0 Sequence $S_1=1\cdot_7 j$ 0 1 2 3 4 5 6 Sequence $S_2=2\cdot_7 j$ 0 2 4 6 1 3 5 Sequence $S_3=3\cdot_7 j$ 0 3 6 2 5 1 4 Sequence $S_4=4\cdot_7 j$ 0 4 1 5 2 6 3 Sequence $S_5=5\cdot_7 j$ 0 5 3 1 6 4 2 Sequence $S_6=6\cdot_7 j$ 0 6 5 4 3 2 1
 $j \in \mathbb{F}_7$ 0 1 2 3 4 5 6 Sequence $S_0=0 \cdot_7 j$ 0 0 0 0 0 0 0 Sequence $S_1=1\cdot_7 j$ 0 1 2 3 4 5 6 Sequence $S_2=2\cdot_7 j$ 0 2 4 6 1 3 5 Sequence $S_3=3\cdot_7 j$ 0 3 6 2 5 1 4 Sequence $S_4=4\cdot_7 j$ 0 4 1 5 2 6 3 Sequence $S_5=5\cdot_7 j$ 0 5 3 1 6 4 2 Sequence $S_6=6\cdot_7 j$ 0 6 5 4 3 2 1
Set of HMC sequences for $p = 7$ with minimum distance $d$ between consecutive elements
 Sequence $H_1$ 1 3 5 7 9 11 6 $d=2$ Sequence $H_2$ 2 6 10 7 4 8 5 $d=3$ Sequence $H_3$ 3 9 8 7 6 5 4 $d=1$ Sequence $H_4$ 4 5 6 7 8 9 3 $d=1$ Sequence $H_5$ 5 8 4 7 10 6 2 $d=3$ Sequence $H_6$ 6 11 9 7 5 3 1 $d=2$
 Sequence $H_1$ 1 3 5 7 9 11 6 $d=2$ Sequence $H_2$ 2 6 10 7 4 8 5 $d=3$ Sequence $H_3$ 3 9 8 7 6 5 4 $d=1$ Sequence $H_4$ 4 5 6 7 8 9 3 $d=1$ Sequence $H_5$ 5 8 4 7 10 6 2 $d=3$ Sequence $H_6$ 6 11 9 7 5 3 1 $d=2$
The set of prime sequences $S_i$, $1 \leq i \leq 18$ for $p = 19$
 $S_1 =$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 $S_2 =$ 0 2 4 6 8 10 12 14 16 18 1 3 5 7 9 11 13 15 17 $S_3 =$ 0 3 6 9 12 15 18 2 5 8 11 14 17 1 4 7 10 13 16 $S_4 =$ 0 4 8 12 16 1 5 9 13 17 2 6 10 14 18 3 7 11 15 $S_5 =$ 0 5 10 15 1 6 11 16 2 7 12 17 3 8 13 18 4 9 14 $S_6 =$ 0 6 12 18 5 11 17 4 10 16 3 9 15 2 8 14 1 7 13 $S_7 =$ 0 7 14 2 9 16 4 11 18 6 13 1 8 15 3 10 17 5 12 $S_8 =$ 0 8 16 5 13 2 10 18 7 15 4 12 1 9 17 6 14 3 11 $S_9 =$ 0 9 18 8 17 7 16 6 15 5 14 4 13 3 12 2 11 1 10 $S_{10} =$ 0 10 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 $S_{11} =$ 0 11 3 14 6 17 9 1 12 4 15 7 18 10 2 13 5 16 8 $S_{12} =$ 0 12 5 17 10 3 15 8 1 13 6 18 11 4 16 9 2 14 7 $S_{13} =$ 0 13 7 1 14 8 2 15 9 3 16 10 4 17 11 5 18 12 6 $S_{14} =$ 0 14 9 4 18 13 8 3 17 12 7 2 16 11 6 1 15 10 5 $S_{15} =$ 0 15 11 7 3 18 14 10 6 2 17 13 9 5 1 16 12 8 4 $S_{16} =$ 0 16 13 10 7 4 1 17 14 11 8 5 2 18 15 12 9 6 3 $S_{17} =$ 0 17 15 13 11 9 7 5 3 1 18 16 14 12 10 8 6 4 2 $S_{18} =$ 0 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
 $S_1 =$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 $S_2 =$ 0 2 4 6 8 10 12 14 16 18 1 3 5 7 9 11 13 15 17 $S_3 =$ 0 3 6 9 12 15 18 2 5 8 11 14 17 1 4 7 10 13 16 $S_4 =$ 0 4 8 12 16 1 5 9 13 17 2 6 10 14 18 3 7 11 15 $S_5 =$ 0 5 10 15 1 6 11 16 2 7 12 17 3 8 13 18 4 9 14 $S_6 =$ 0 6 12 18 5 11 17 4 10 16 3 9 15 2 8 14 1 7 13 $S_7 =$ 0 7 14 2 9 16 4 11 18 6 13 1 8 15 3 10 17 5 12 $S_8 =$ 0 8 16 5 13 2 10 18 7 15 4 12 1 9 17 6 14 3 11 $S_9 =$ 0 9 18 8 17 7 16 6 15 5 14 4 13 3 12 2 11 1 10 $S_{10} =$ 0 10 1 11 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 $S_{11} =$ 0 11 3 14 6 17 9 1 12 4 15 7 18 10 2 13 5 16 8 $S_{12} =$ 0 12 5 17 10 3 15 8 1 13 6 18 11 4 16 9 2 14 7 $S_{13} =$ 0 13 7 1 14 8 2 15 9 3 16 10 4 17 11 5 18 12 6 $S_{14} =$ 0 14 9 4 18 13 8 3 17 12 7 2 16 11 6 1 15 10 5 $S_{15} =$ 0 15 11 7 3 18 14 10 6 2 17 13 9 5 1 16 12 8 4 $S_{16} =$ 0 16 13 10 7 4 1 17 14 11 8 5 2 18 15 12 9 6 3 $S_{17} =$ 0 17 15 13 11 9 7 5 3 1 18 16 14 12 10 8 6 4 2 $S_{18} =$ 0 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
The set of HMC sequences $H_i$, $1 \leq i \leq 18$ for $p = 19$ with minimum distance $d$ between consecutive elements
 $H_1 =$ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 18 $d=2$ $H_2 =$ 2 6 10 14 18 22 26 30 34 19 4 8 12 16 20 24 28 32 17 $d = 4$ $H_3 =$ 3 9 15 21 27 33 20 7 13 19 25 31 18 5 11 17 23 29 16 $d = 6$ $H_4 =$ 4 12 20 28 17 6 14 22 30 19 8 16 24 32 21 10 18 26 15 $d = 8$ $H_5 =$ 5 15 25 16 7 17 27 18 9 19 29 20 11 21 31 22 13 23 14 $d = 9$ $H_6 =$ 6 18 30 23 16 28 21 14 26 19 12 24 17 10 22 15 8 20 13 $d = 7$ $H_7 =$ 7 21 16 11 25 20 15 29 24 19 14 9 23 18 13 27 22 17 12 $d = 5$ $H_8 =$ 8 24 21 18 15 12 28 25 22 19 16 13 10 26 23 20 17 14 11 $d = 3$ $H_9 =$ 9 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 $d = 1$ $H_{10} =$ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 9 $d = 1$ $H_{11} =$ 11 14 17 20 23 26 10 13 16 19 22 25 28 12 15 18 21 24 8 $d = 3$ $H_{12} =$ 12 17 22 27 13 18 23 9 14 19 24 29 15 20 25 11 16 21 7 $d = 5$ $H_{13} =$ 13 20 8 15 22 10 17 24 12 19 26 14 21 28 16 21 30 18 6 $d = 7$ $H_{14} =$ 14 23 13 22 31 21 11 20 29 19 9 18 27 17 7 16 25 15 5 $d = 9$ $H_{15} =$ 15 26 18 10 21 32 24 16 8 19 30 22 14 5 17 28 20 12 4 $d = 8$ $H_{16} =$ 16 29 23 17 11 5 18 31 25 19 13 7 20 33 27 21 15 9 3 $d = 6$ $H_{17} =$ 17 32 28 24 20 16 12 8 4 19 34 30 26 22 18 14 10 6 2 $d = 4$ $H_{18} =$ 18 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1 $d = 2$
 $H_1 =$ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 18 $d=2$ $H_2 =$ 2 6 10 14 18 22 26 30 34 19 4 8 12 16 20 24 28 32 17 $d = 4$ $H_3 =$ 3 9 15 21 27 33 20 7 13 19 25 31 18 5 11 17 23 29 16 $d = 6$ $H_4 =$ 4 12 20 28 17 6 14 22 30 19 8 16 24 32 21 10 18 26 15 $d = 8$ $H_5 =$ 5 15 25 16 7 17 27 18 9 19 29 20 11 21 31 22 13 23 14 $d = 9$ $H_6 =$ 6 18 30 23 16 28 21 14 26 19 12 24 17 10 22 15 8 20 13 $d = 7$ $H_7 =$ 7 21 16 11 25 20 15 29 24 19 14 9 23 18 13 27 22 17 12 $d = 5$ $H_8 =$ 8 24 21 18 15 12 28 25 22 19 16 13 10 26 23 20 17 14 11 $d = 3$ $H_9 =$ 9 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 $d = 1$ $H_{10} =$ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 9 $d = 1$ $H_{11} =$ 11 14 17 20 23 26 10 13 16 19 22 25 28 12 15 18 21 24 8 $d = 3$ $H_{12} =$ 12 17 22 27 13 18 23 9 14 19 24 29 15 20 25 11 16 21 7 $d = 5$ $H_{13} =$ 13 20 8 15 22 10 17 24 12 19 26 14 21 28 16 21 30 18 6 $d = 7$ $H_{14} =$ 14 23 13 22 31 21 11 20 29 19 9 18 27 17 7 16 25 15 5 $d = 9$ $H_{15} =$ 15 26 18 10 21 32 24 16 8 19 30 22 14 5 17 28 20 12 4 $d = 8$ $H_{16} =$ 16 29 23 17 11 5 18 31 25 19 13 7 20 33 27 21 15 9 3 $d = 6$ $H_{17} =$ 17 32 28 24 20 16 12 8 4 19 34 30 26 22 18 14 10 6 2 $d = 4$ $H_{18} =$ 18 35 33 31 29 27 25 23 21 19 17 15 13 11 9 7 5 3 1 $d = 2$
Set of 14 HMC sequences of period 19 and adjacent distance $\geq 3$
 $H_2 =$ 2 6 10 14 18 22 26 30 34 19 4 8 12 16 20 24 28 32 17 $d = 4$ $H_3 =$ 3 9 15 21 27 33 20 7 13 19 25 31 18 5 11 17 23 29 16 $d = 6$ $H_4 =$ 4 12 20 28 17 6 14 22 30 19 8 16 24 32 21 10 18 26 15 $d = 8$ $H_5 =$ 5 15 25 16 7 17 27 18 9 19 29 20 11 21 31 22 13 23 14 $d = 9$ $H_6 =$ 6 18 30 23 16 28 21 14 26 19 12 24 17 10 22 15 8 20 13 $d = 7$ $H_7 =$ 7 21 16 11 25 20 15 29 24 19 14 9 23 18 13 27 22 17 12 $d = 5$ $H_8 =$ 8 24 21 18 15 12 28 25 22 19 16 13 10 26 23 20 17 14 11 $d = 3$ $H_{11} =$ 11 14 17 20 23 26 10 13 16 19 22 25 28 12 15 18 21 24 8 $d = 3$ $H_{12} =$ 12 17 22 27 13 18 23 9 14 19 24 29 15 20 25 11 16 21 7 $d = 5$ $H_{13} =$ 13 20 8 15 22 10 17 24 12 19 26 14 21 28 16 21 30 18 6 $d = 7$ $H_{14} =$ 14 23 13 22 31 21 11 20 29 19 9 18 27 17 7 16 25 15 5 $d = 9$ $H_{15} =$ 15 26 18 10 21 32 24 16 8 19 30 22 14 5 17 28 20 12 4 $d = 8$ $H_{16} =$ 16 29 23 17 11 5 18 31 25 19 13 7 20 33 27 21 15 9 3 $d = 6$ $H_{17} =$ 17 32 28 24 20 16 12 8 4 19 34 30 26 22 18 14 10 6 2 $d = 4$
 $H_2 =$ 2 6 10 14 18 22 26 30 34 19 4 8 12 16 20 24 28 32 17 $d = 4$ $H_3 =$ 3 9 15 21 27 33 20 7 13 19 25 31 18 5 11 17 23 29 16 $d = 6$ $H_4 =$ 4 12 20 28 17 6 14 22 30 19 8 16 24 32 21 10 18 26 15 $d = 8$ $H_5 =$ 5 15 25 16 7 17 27 18 9 19 29 20 11 21 31 22 13 23 14 $d = 9$ $H_6 =$ 6 18 30 23 16 28 21 14 26 19 12 24 17 10 22 15 8 20 13 $d = 7$ $H_7 =$ 7 21 16 11 25 20 15 29 24 19 14 9 23 18 13 27 22 17 12 $d = 5$ $H_8 =$ 8 24 21 18 15 12 28 25 22 19 16 13 10 26 23 20 17 14 11 $d = 3$ $H_{11} =$ 11 14 17 20 23 26 10 13 16 19 22 25 28 12 15 18 21 24 8 $d = 3$ $H_{12} =$ 12 17 22 27 13 18 23 9 14 19 24 29 15 20 25 11 16 21 7 $d = 5$ $H_{13} =$ 13 20 8 15 22 10 17 24 12 19 26 14 21 28 16 21 30 18 6 $d = 7$ $H_{14} =$ 14 23 13 22 31 21 11 20 29 19 9 18 27 17 7 16 25 15 5 $d = 9$ $H_{15} =$ 15 26 18 10 21 32 24 16 8 19 30 22 14 5 17 28 20 12 4 $d = 8$ $H_{16} =$ 16 29 23 17 11 5 18 31 25 19 13 7 20 33 27 21 15 9 3 $d = 6$ $H_{17} =$ 17 32 28 24 20 16 12 8 4 19 34 30 26 22 18 14 10 6 2 $d = 4$
 [1] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [2] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [3] Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300 [4] Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 [5] Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 [6] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [7] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [8] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [9] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [10] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270 [11] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [12] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [13] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

2019 Impact Factor: 0.734