215 research outputs found
Parity properties of Costas arrays defined via finite fields
A Costas array of order is an arrangement of dots and blanks into
rows and columns, with exactly one dot in each row and each column, the
arrangement satisfying certain specified conditions. A dot occurring in such an
array is even/even if it occurs in the -th row and -th column, where
and are both even integers, and there are similar definitions of odd/odd,
even/odd and odd/even dots. Two types of Costas arrays, known as Golomb-Costas
and Welch-Costas arrays, can be defined using finite fields. When is a
power of an odd prime, we enumerate the number of even/even odd/odd, even/odd
and odd/even dots in a Golomb-Costas array. We show that three of these numbers
are equal and they differ by from the fourth. For a Welch-Costas array
of order , where is an odd prime, the four numbers above are all equal
to when , but when , we show
that the four numbers are defined in terms of the class number of the imaginary
quadratic field , and thus behave in a much less
predictable manner.Comment: To appear in Advances in Mathematics of Communication
Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes
A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid
A review of Costas arrays
Costas arrays are not only useful in radar engineering, but they
also present many interesting, and still open, mathematical
problems. This work collects in it all important knowledge about
them available today: some history of the subjects, density
results, construction methods, construction algorithms with full
proofs, and open questions. At the same time all the necessary
mathematical background is offered in the simplest possible format
and terms, so that this work can play the role of a reference for
mathematicians and mathematically inclined engineers interested in the field
Algebraic symmetries of generic dimensional periodic Costas arrays
In this work we present two generators for the group of symmetries of the
generic dimensional periodic Costas arrays over elementary abelian
groups: one that is defined by multiplication on
dimensions and the other by shear (addition) on dimensions. Through
exhaustive search we observe that these two generators characterize the group
of symmetries for the examples we were able to compute. Following the results,
we conjecture that these generators characterize the group of symmetries of the
generic dimensional periodic Costas arrays over elementary abelian
groups
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