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

Lenny Fukshansky; Ahmad A. Shaar

doi:10.3934/amc.2018012

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2018, Volume 12, Issue 1: 181-188. Doi: 10.3934/amc.2018012

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

Lenny Fukshansky^1, and
Ahmad A. Shaar^2,

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: February 2018

Abstract / Introduction Full Text(HTML) Figure(0) / Table(5) Related Papers Cited by

Abstract

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.

Keywords:

Frequency hopping,
Hamming correlation property,
one-coincidence sequence sets with dispersed sequence elements.

Mathematics Subject Classification: Primary: 94A12, 94A15, 94.10; Secondary: 11B50.

Citation:

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Table 1. Set of prime sequences for $p = 7$

$j \in \mathbb{F}_7$	1	2	3	4	5	6
Sequence $S_0=0 \cdot_7 j$	0	0	0	0	0	0
Sequence $S_1=1\cdot_7 j$	1	2	3	4	5	6
Sequence $S_2=2\cdot_7 j$	2	4	6	1	3	5
Sequence $S_3=3\cdot_7 j$	3	6	2	5	1	4
Sequence $S_4=4\cdot_7 j$	4	1	5	2	6	3
Sequence $S_5=5\cdot_7 j$	5	3	1	6	4	2
Sequence $S_6=6\cdot_7 j$	6	5	4	3	2	1

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Table 2. 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	$d=2$

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Table 3. The set of prime sequences $S_i$, $1 \leq i \leq 18$ for $p = 19$

$S_1 = $	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
$S_2 = $	2	4	6	8	10	12	14	16	18	1	3	5	7	9	11	13	15	17
$S_3 = $	3	6	9	12	15	18	2	5	8	11	14	17	1	4	7	10	13	16
$S_4 = $	4	8	12	16	1	5	9	13	17	2	6	10	14	18	3	7	11	15
$S_5 = $	5	10	15	1	6	11	16	2	7	12	17	3	8	13	18	4	9	14
$S_6 = $	6	12	18	5	11	17	4	10	16	3	9	15	2	8	14	1	7	13
$S_7 = $	7	14	2	9	16	4	11	18	6	13	1	8	15	3	10	17	5	12
$S_8 = $	8	16	5	13	2	10	18	7	15	4	12	1	9	17	6	14	3	11
$S_9 = $	9	18	8	17	7	16	6	15	5	14	4	13	3	12	2	11	1	10
$S_{10} = $	10	1	11	2	12	3	13	4	14	5	15	6	16	7	17	8	18	9
$S_{11} = $	11	3	14	6	17	9	1	12	4	15	7	18	10	2	13	5	16	8
$S_{12} = $	12	5	17	10	3	15	8	1	13	6	18	11	4	16	9	2	14	7
$S_{13} = $	13	7	1	14	8	2	15	9	3	16	10	4	17	11	5	18	12	6
$S_{14} = $	14	9	4	18	13	8	3	17	12	7	2	16	11	6	1	15	10	5
$S_{15} = $	15	11	7	3	18	14	10	6	2	17	13	9	5	1	16	12	8	4
$S_{16} = $	16	13	10	7	4	1	17	14	11	8	5	2	18	15	12	9	6	3
$S_{17} = $	17	15	13	11	9	7	5	3	1	18	16	14	12	10	8	6	4	2
$S_{18} = $	18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1

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Table 4. 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	$d = 2$

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Table 5. 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	$d = 4$

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

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