Costas array
of order $ n $
An $ ( n \times n ) $ permutation matrix in which the $ { {n ( n  1 ) } / 2 } $ line segments connecting pairs of ones in the matrix are distinct as vectors, i.e., no two agree in both magnitude and slope. Thus, if $ a _ {ij } = a _ {kl } = 1 $ and $ a _ {mn } = a _ {pq } = 1 $ in the matrix, with $ ( i,j ) \neq ( k,l ) $, $ ( m,n ) \neq ( p,q ) $, and at least three of these four ordered pairs are distinct, and if $  {( a _ {ij } ,a _ {kl } ) }  ^ {2} = ( k  i ) ^ {2} + ( l  j ) ^ {2} $ equals $  {( a _ {mn } ,a _ {pq } ) }  ^ {2} = ( p  m ) ^ {2} + ( q  n ) ^ {2} $, then
$$ \left  { { \frac{( l  j ) }{( k  i ) } } } \right  \neq \left  { { \frac{( q  n ) }{( p  m ) } } } \right  . $$
All known (1996) general constructions for Costas arrays (see [a3], [a4], [a5]) involve primitive roots in finite fields (cf. Field). The Welch construction, for every prime $ p > 2 $, gives a Costas array of order $ p  1 $ by setting $ a _ {ij } = 1 $ when $ j = g ^ {i} ( { \mathop{\rm mod} } p ) $, where $ g $ is a primitive root modulo $ p $. Removing row $ p  1 $ and column $ 1 $ from this construction leaves a Costas array of order $ p  2 $. When $ g = 2 $ is used as the primitive root modulo $ p $, the array can be further reduced to order $ p  3 $.
In Golomb's construction, if $ \alpha $ and $ \beta $ are any two primitive elements in $ { \mathop{\rm GF} } ( q ) $, for $ q > 2 $, a Costas array of order $ q  2 $ is obtained by setting $ a _ {ij } = 1 $ whenever $ \alpha ^ {i} + \beta ^ {j} = 1 $. (The case $ \alpha = \beta $ had been discovered by A. Lempel.) If $ \alpha + \beta = 1 $, then $ a _ {11 } = 1 $, so that by removing the first row and first column, a Costas array or order $ q  3 $ is obtained. As shown in [a6], for every $ q > 2 $ one can find primitive roots $ \alpha $ and $ \beta $( not necessarily distinct) with $ \alpha + \beta = 1 $.
For $ q = 4,5,9 $, or any prime $ p \equiv \pm 1 ( { \mathop{\rm mod} } 10 ) $, and for no other finite fields, there is a primitive root $ \alpha $ with $ \alpha + \alpha ^ {2} = 1 $. Thus, in the Lempel construction, $ a _ {12 } = a _ {21 } = 1 $, so that if the first two rows and first two columns are removed, a (symmetric) Costas array or order $ q  4 $ results. For $ q = 4,5,9 $, or any prime $ p \equiv 1,9 ( { \mathop{\rm mod} } 20 ) $, there are primitive roots $ \alpha $ and $ \beta $ with $ \alpha + \beta = 1 $ and $ \alpha ^ {2} + \beta ^ { 1 } = 1 $. In this case, $ a _ {11 } = 1 $, $ a _ {2,q  2 } = 1 $ and $ a _ {q  2,2 } = 1 $. By successive removal of rows and columns Costas arrays of orders $ q  2 $, $ q  3 $, $ q  4 $, and $ q  5 $ are obtained for these values of $ q $.
In some cases a Costas array of order $ n + 1 $ can be obtained from one of order $ n $ by adjoining a $ 1 $ in an exterior corner, i.e., in one of the positions $ a _ {00 } $, $ a _ {0,n + 1 } $, $ a _ {n + 1,0 } $, or $ a _ {n + 1,n + 1 } $. Several primeorder examples were obtained in this way from a Welch example of order $ p  1 $.
The total number $ C ( n ) $ of Costas arrays of order $ n $ is known (1996) for $ n \leq 23 $( see the table), with a local maximum at $ n = 16 $.
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For a prime number $ p $ there are $ \phi ( p  1 ) $ primitive roots, each of which leads to a different Costas array of order $ p  1 $. Since $ \phi ( p  1 ) $ can be arbitrarily large, $ {\lim\limits \sup } _ {n \rightarrow \infty } C ( n ) = \infty $. It has been conjectured, but not proved, that $ {\lim\limits \inf } _ {n \rightarrow \infty } C ( n ) = 0 $. This would require $ C ( n ) = 0 $ infinitely often. No case of $ C ( n ) = 0 $ is yet known, but no examples of Costas arrays of orders $ 32 $, $ 33 $, or $ 43 $ have yet been found. (Many larger orders also lack examples.)
J.P. Costas [a1] first proposed these arrays for an application to frequency hopping sonar signals. Let the $ n $ rows represent $ n $ equally spaced frequencies, and the $ n $ columns $ n $ equal duration time intervals. Then the Costas array specifies a permuted order for the $ n $ frequencies to be transmitted in $ n $ consecutive time intervals. As a sonar (or radar) signal this design has an ideal "thumbtack" ambiguity function (the twodimensional autocorrelation function in time and frequency). The horizontal (time) shift measures range (the distance to the target) and the vertical (frequency) shift measures the Doppler (the velocity of the target relative to the observer). For any nonzero shift parallel to the coordinate axes a Costas array has at most one "hit" (coincidence of a $ 1 $ with a $ 1 $), and thus gives the least ambiguous reading of the correct range and Doppler in the presence of noise.
References
[a1]  J.P. Costas, "Project medior. A mediumoriented approach to sonar signal processing" , HMED Tech. Publ. R66EMH12 , General Electric (1966) 
[a2]  S.W. Golomb, H. Taylor, "Twodimensional synchronization patterns with minimum ambiguity" IEEE Trans. Inform. Theory , 28 (1982) pp. 600–604 
[a3]  S.W. Golomb, H. Taylor, "Constructions and properties of Costas arrays" Proc. IEEE , 72 (1984) pp. 1143–1163 
[a4]  S.W. Golomb, "Algebraic constructions for Costas arrays" J. Combin. Th. A , 37 (1984) pp. 13–21 
[a5]  S.W. Golomb, "The and constructions for Costas arrays" IEEE Trans. Inform. Th. , 38 (1992) pp. 1404–1406 
[a6]  O. Moreno, J. Sotero, "Computational approach to conjecture A of Golomb" Congressus Numerantium , 70 (1990) pp. 7–16 
[a7]  J.P. Costas, "A study of a class of detection waveforms having nearly ideal rangedoppler ambiguity properties" Proc. IEEE , 72 (1984) pp. 996–1009 
Costas array. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Costas_array&oldid=46534