# 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. [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. [3] C. Hodgdon, Adaptive frequency hopping for reduced interference between bluetooth and wireless LAN, Ericsson Technology Licensing, 2003. [4] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [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.

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##### 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. [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. [3] C. Hodgdon, Adaptive frequency hopping for reduced interference between bluetooth and wireless LAN, Ericsson Technology Licensing, 2003. [4] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [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.
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 $d = 4$
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