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examples for various values of
m
by carefully examining the back-track algorithm
for the search.
From this estimate, we could have concluded that no full-size modular sonar
sequence exists for
m
beyond a certain value. This is, however, not true, since
there are some algebraic constructions which give full size examples (of length
m
+1 on
m
symbols) for infinite values of
m
. We could safely guess that any full-
size example for large values of
m
must be either from an algebraic construction,
or else the probability that it exists is extremely small.
We still leave the following problems unsolved:
Unsolved Problem 1.
Find an example of 35
×
35 modular sonar sequences
(mod 35) or prove that none exists.
Unsolved Problem 2.
Generalize the above to the case of
m
=
p
(
p
+2) being
a product of twin primes.
Unsolved Problem 3.
Find infinitely many values of
m
for which an
m
×
(
m
+
1) modular sonar sequences do not exist.
Unsolved Problem 4.
Except for
m
being a prime or one less than a prime
power, would the fact that the value in (6) is close to zero imply non-
existence?
Unsolved Problem 5.
How accurate is the estimate in (6)?
Unsolved Problem 6.
Could a similar approach be used to estimate the num-
ber of Costas arrays, see [7] ?
References
1. Costas, J.P.: Medium constraints on sonar design and performance. FASCON
CONV. Rec. 68A-68L (1975)
2. Erdos, P., Graham, R.L., Ruzsa, I.Z., Taylor, H.: Bounds for arrays of dots with
distinct slopes or lengths. Combinatorica 12, 1-6 (1992)
3. Moreno, O., Games, R.A., Taylor, H.: Sonar sequences from costas arrays and the
best known sonar sequences with up to 100 symbols. IEEE Trans. Inform. The-
ory 39(6) (November 1987)
4. Gagliardi, R., Robbins, J., Taylor, H.: Acquisition sequences in PPM communica-
tions. IEEE Trans. Inform. Theory IT-33, 738-744 (1987)
5. Games, R.A.: An algebraic construction of sonar sequences using M-sequences.
SIAM J. Algebraic Discrete Methods 8, 753-761 (1987)
6. Golomb, S.W., Gong, G.: Signal Design for Good Correlation. Cambridge University
Press, Cambridge (2005)
7. Golomb, S.W., Taylor, H.: Two-dimensional synchronization patterns for minimum
ambiguity. IEEE Trans. Inform. Theory IT-28, 263-272 (1982)
8. Yoon., S.-J., Song, H.-Y.: Existence of Modular Sonar Sequences of Twin-Prime
Product Length. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) SSC
2007. LNCS, vol. 4893, pp. 184-191. Springer, Heidelberg (2007)
9. Silverman, J., Vickers, V.E., Mooney, J.M.: On the number of Costas arrays as a
function of array size. Proceedings of the IEEE 76(7) (July 1988)
