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A Probabilistic Approach on Estimating the
Number of Modular Sonar Sequences
Ki-Hyeon Park and Hong-Yeop Song
Department of Electrical and Electronic Engineering
Yonsei University, Seoul, 121-749, Korea
{
kh.park, hysong
}
@yonsei.ac.kr
Abstract.
We report some results of an extensive computer search for
m×n
modular sonar sequences and estimate the number of inequivalent
examples of size
m×n
using a probabilistic approach. Evidence indicates
strongly that a full size example exists with extremely small probability
for large
m
.
1
Introduction
A sonar sequence is an integer sequence that has some interesting properties for
use in communication applications. Its mathematical concept was well described
in [2] and the original motivation and application to some communication prob-
lems can be found in [1,6].
Recently, [8] has discussed a search for a 35
×
35 modular sonar sequence and,
in general,
m
m
examples where
m
=
p
(
p
+2) is a product of twin primes. It was
for their application to the design of CDMA sequences, but they failed to find
any single example beyond
m
= 15. This paper is an attempt to continue this
effort, and shows some results of an extensive computer search for small values
of
m
. Based on the search result, we now believe that no
m
×
(
m
+ 1) modular
sonar sequence exists, except for those given by the algebraic constructions. To
explain this, we use some probabilistic approaches for estimating the number of
m
×
×
(
m
+ 1) modular sonar sequences.
An
m
×
n
sonar sequence is defined as a function from the set of integers
=
A
n
to the set of integers
=
A
m
with the following
{
1
,
2
, ..., n
}
{
1
,
2
, ..., m
}
distinct difference property (DDP) [3].
Definition 1.
(DDP) A function
f
:
A
n
→
A
m
has a distinct difference prop-
erty if for all integers
h
,
i
,and
j
,with
1
≤
h
≤
n
−
1
and
1
≤
i
,
j
≤
n
−
h
,
f
(
i
+
h
)
−
f
(
i
)=
f
(
j
+
h
)
−
f
(
j
)
implies
i
=
j.
(1)
An
m
A
m
with DDP. The main
problem in sonar sequences research is to determine the maximum value
n
for
each given
m
such that an
m
×
n
sonar sequence is a function
f
:
A
n
→
n
sonar sequence exists. For values of
m
up
to 100, the best known
n
is reported in [3]. To obtain these values, they have
introduced “modular sonar sequences.” A modular sonar sequence is a sonar
×
