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2000 Mathematics Subject Classification: 11B50, 11B75, 05B10. Key words and phrases: Costas arrays, Golomb construction, Welch construction. All authors ...

Advances in Mathematics of Communications Volume 3, No. 1, 2009, 35–52

doi:10.3934/amc.2009.3.35

COMMON DISTANCE VECTORS BETWEEN COSTAS ARRAYS

Konstantinos Drakakis School of Electrical, Electronic & Mechanical Engineering University College Dublin, Belfield, Dublin 4, Ireland

Roderick Gow School of Mathematics University College Dublin, Belfield, Dublin 4, Ireland

Scott Rickard School of Electrical, Electronic & Mechanical Engineering University College Dublin, Belfield, Dublin 4, Ireland

(Communicated by Andrew Klapper) Abstract. We investigate the distance vectors contained in individual Costas arrays and in pairs of Costas arrays, and prove some rigorous results in the case of the algebraically constructed arrays. Overall, it appears that the set with the property that every Costas array has a distance vector in this set, or that every pair of Costas arrays with a common vector have a common vector in this set, is in both cases surprisingly small. Further, we study Costas arrays with the additional property that they represent configurations of nonattacking kings or queens: in the former case, we demonstrate that such arrays are either sporadic or produced by a sub-method of the Lempel construction; in the latter case, partially answering a question asked by S. Golomb 26 years ago, we prove that (non-trivial) such arrays can only be sporadic and conjecture they do not exist at all.

1. Introduction Costas arrays/permutations [1, 2, 3] have numerous applications in RADAR and SONAR systems, time synchronization etc. They have been the object of active research for more than 40 years now [1], but, on the theoretical front, there has been a stalemate for the last 20 years or so, ever since the publication of the known algebraic construction methods by Golomb et al. in 1984 [8]. Among the various semi-empirical methods suggested for the construction of new Costas arrays was “interlacing” two Costas arrays of the same order to produce a Costas array of order twice as high; it was shown [7], however, that any two Costas arrays of equal orders have a distance vector in common (with the exception of the trivially small orders 1 and 2), implying that the interlaced array would never be Costas. The same result (except again for very small orders) holds for any two Costas arrays of orders differing by 1 [5], where interlacing is still technically possible. For two Costas arrays of orders differing by 2 or more it is not possible to extend the interlacing method, at least in an obvious way. It still makes sense, however, 2000 Mathematics Subject Classification: 11B50, 11B75, 05B10. Key words and phrases: Costas arrays, Golomb construction, Welch construction. All authors are also affiliated with the Claude Shannon Institute (www.shannoninstitute.ie ), as well as with UCD CASL (casl.ucd.ie). 35

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to investigate whether it remains true that any such pair of Costas arrays have a vector in common. The question is relevant in the context of wireless multiuser communications, where multiple Costas frequency-hopping waveforms used in the same location can lead to cross-talk, whether they be of unequal orders or not. Experimental evidence strongly suggests that most of the time two Costas arrays do have a vector in common. What would be then the smallest set of distance vectors with the property that any pair of Costas arrays with a distance vector in common have a common vector lying in that set? Similarly, what is the smallest set of distance vectors with the property that any Costas array has a distance vector lying in this set? In other words, how diverse is the pool of (common) distance vectors of Costas arrays? The closest two dots of a Costas array can lie is in adjacent rows and columns, √ hence the shortest distance vectors possible are (1, 1) and (1, −1), of length 2. As they can potentially exist in all Costas arrays of order larger than 1, we would expect them to be contained in the sets mentioned above, and perhaps very nearly essentially “be” the sets by themselves, in the sense that most pairs of Costas arrays with a vector in common will have one of these two vectors in common, and that most Costas arrays will contain one of these two vectors. We propose then to pursue the question of which Costas arrays contain such a vector as yet another problem in its own sake. A quick pass over the database of known Costas arrays (of order at most 27) reveals that Costas arrays not containing either (1, 1) or (1, −1) exist but are very few; geometrically, this property implies that, for each dot in the array, the neighboring eight squares (or five, if the dot lies on an edge of the array, or even three, if it lies at a corner) are dot-free. If we imagine the dots to be kings and the array to be a chessboard, then this is clearly a configuration of non-attacking kings. In this work we study these special Costas arrays, named non-attacking kings Costas arrays (NAKCAs∗ ), and we also describe which subcategories of Costas arrays we should seek them in. We also see that we can widen in a meaningful way the set of distance vectors that we would not like a Costas array to contain, and still get Costas arrays that satisfy the condition; the samples we get are, however, too few to draw any meaningful conclusions regarding their existence and construction. Finally, having established a connection between Costas arrays and the chessboard problem of non-attacking kings, we could not possibly miss the opportunity to investigate a link with the far more celebrated problem of non-attacking queens. In this new context, however, we fail to be as assertive as before: we can prove that all non-trivial examples of such Costas arrays must be sporadic, but we fail to locate even one such example, and this arouses the suspicion that such arrays do not exist at all. 2. Basics Simply put, a Costas array is a square arrangement of dots and blanks, such that there is exactly one dot per row and column, and such that all vectors between dots are distinct. Definition 1. Let f : [n] → [n], where [n] = {1, . . . , n}, n ∈ N, be a bijection; then f has the Costas property iff the collection of vectors {(i − j, f (i) − f (j)) : 1 ≤ j < i ≤ n}, called the distance vectors, are all distinct, in which case it is called ∗ The

word “n¯ aqa”, sounding the same as NAKCA, means “female camel” in Arabic.

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a Costas permutation. The corresponding Costas array is the square array n × n where the elements at (f (i), i), i ∈ [n] are equal to 1 (dots), while the remaining elements are equal to 0 (blanks). Remark 1. • The operations of horizontal flip, vertical flip, and transposition on a Costas array result to a Costas array as well: hence, out of a Costas array 8 can be created, or 4 if the particular Costas array is symmetric. • A Costas array is actually a NAKCA iff neither (1, −1) nor (1, 1) are among the distance vectors. Remark 2. The unusual convention whereby the order of the indices in the definition of array element positions is inverted with respect to the definition of distance vectors was adopted so that both definitions remain natural: in array descriptions the first index describes the vertical position of an element and the second the horizontal, whereas vectors are naturally interpreted on a plane, where the first coordinate shows the horizontal component and the second the vertical. Note though that positive vertical (second) distance vector coordinates point downwards in our definition, not upwards, so that we avoid further confusion through the introduction of an extra minus sign. The horizontal (first) component of a distance vector is chosen by convention to be always positive, so we always consider vectors pointing from left to right. As an example, the distance vector between the elements a14 and a35 of an array A is (5 − 4, 3 − 1) = (1, 2): the second coordinate is positive, so the vector should point from top left to bottom right (namely downwards), as indeed it is easily verified to do. In the sequel we will make no distinction between the Costas array and the corresponding permutation. Definition 2. Let f : [n] → [n] be a permutation; its difference triangle T (f ) is the collection of multisets ti (f ) = {f (k) − f (i + k) : k ∈ [n − i]}, i ∈ [n − 1], called the rows of the triangle. Remark 3. • The Costas property is equivalent to the fact that no row of the difference triangle contains a given entry more than once; in other words, the rows are sets rather than multisets. The proof is simple: row i contains the second coordinates of those distance vectors whose first coordinate is equal to i ∈ [n−1], hence a duplicate entry in a row would immediately imply the existence of two equal distance vectors, in violation of the Costas property. • Note that a Costas array is actually a NAKCA iff t1 (f ) contains no entry of absolute value 1. There are essentially two construction methods for Costas arrays, based on the algebraic theory of finite fields, each of which has a number of possible extensions, some applicable systematically and some haphazardly. Let us see them without proof: Theorem 1 (Welch exponential construction W1 ). Let p be prime and g a primitive root of the field F(p); for c ∈ {0, . . . , p − 2} constant, the permutation f (i) = g i−1+c

mod p, i = 1, . . . , p − 1

has the Costas property and corresponds to a Costas array of order p − 1. Advances in Mathematics of Communications

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• When c = 0, f (1) = 1: then, by removing f (1), and, setting h(i) = f (i + 1) − 1, i ∈ [p − 2], we create a new Costas permutation h. This is method W2 and is always applicable. • If, in addition, we use g = 2 (this is not always possible), then h(1) = f (2) = 2, and, setting s(i) = h(i+1)−1, i ∈ [p−3], we create a new Costas permutation s. This is method W3 . • It can be shown [4] that the set of exponential Welch arrays created by method W1 is closed under horizontal and vertical flips; however, the set of the transposes of these arrays is completely disjoint from the original set if p > 5: the transposes are called logarithmic Welch arrays. Theorem 2 (Golomb construction G2 ). Let p be a prime, m ∈ N, q = pm and a, b primitive roots of the field F(q); we build the permutation f such that ai + bf (i) = 1, i = 1, . . . , q − 2

corresponding to a Costas array of order q − 2.

• It can be shown that in every finite field there exist two primitive roots a and b such that a + b = 1, in which case G2 yields f (1) = 1, and setting h(i) = f (i + 1) − 1, i ∈ [q − 3], we create a new Costas permutation h. This is method G3 and is always applicable. • When the characteristic of the field is 2 (p = 2), a + b = 1 ⇒ 1 = (a + b)2 = a2 + b2 ⇒ f (2) = 2, and setting s(i) = h(i + 1) − 1, i ∈ [q − 4], we create a new Costas permutation s. This is method G4 . • In the special case where a = b (Lempel case), it may be possible to find a primitive root a such that a2 + a = 1, in which case G2 yields f (1) = 2 and f (2) = 1, so setting t(i) = f (i + 2) − 2, i ∈ [q − 4], we create a new Costas permutation t. This is method T4 . Costas arrays not constructed by the two algebraic methods or their derived methods are commonly referred to as sporadic. 3. Common vectors We wish to find (at least partial) answers to the following problems: Problem 1. Is it true that there exists a n ∈ N, such that for all n1 , n2 ≥ n, any two Costas arrays, one of order n1 and one of order n2 , have a distance vector in common? Problem 2. What is the smallest set S of distance vectors with the property that any pair of Costas arrays (of any, possibly not the same, order) with a distance vector in common have a common distance vector lying in S? Problem 3. What is the smallest set S of distance vectors with the property that any Costas array of order larger than 1 has a distance vector lying in S? We ran an exhaustive search on the database of all known Costas arrays up to order 27 to find pairs without common vectors. Let us denote by C(i, j), i, j ∈ [27] the number of pairs of Costas arrays without common vectors when one array is of order i and the other of order j; obviously, C(i, j) = C(j, i), so we can assume that i ≤ j. The experiment was coded in Matlab and needed approximately 27.8 hours to complete on a Core 2 Duo 6600 (2.4 GHz) PC with 2 GB of memory. The results were as follows: Advances in Mathematics of Communications

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Common distance vectors

i

i

2 3

2 3

2 2

3 4 8

16 7824 424

4 8 0

5 28 0

17 6688 320

6 68 0 18 5344 200

7 104 8 19 3552 136

39

j 9 10 11 12 13 14 15 304 856 1900 3236 5076 6484 7320 8 80 208 176 240 336 432 j 20 21 22 23 24 25 26 27 2196 1208 656 288 32 16 24 40 160 80 16 8 0 0 0 0

8 236 8

Table 1. C(i, j), as defined in Section 3, for i = 2, 3 and j = 2, . . . , 27. • When i = 1, C(1, j) is trivially the total number of Costas arrays of order j, as 1 contains no distance vectors whatsoever! • When i = 2 or i = 3, the results are given in Table 1. • When i = 4, C(4, j) = 0, j ∈ [27], j ≥ 4, j 6= 13, 17, while C(4, 13) = 8 and C(4, 17) = 16. • When 5 ≤ i ≤ j, C(i, j) = 0. This result allows us to formulate a conjecture regarding Problem 1: Conjecture 1. Problem 1 can be answered in the affirmative with n = 5. Let us now turn our attention to Problem 3 for Costas arrays of order n ∈ [27], n ≥ 2. Again, exhaustive search reveals that: • When S = {(1, −1), (1, 1)}, the arrays that contain no distance vector in S are the NAKCAs; Table 2 contains all NAKCAs for the various orders n ≤ 27. • When S = {(1, −1), (1, 1), (1, −2), (1, 2), (2, −1), (2, 1)}, there exist Costas arrays that contain no distance vectors in S, but they are extremely few: our exhaustive search revealed 4 in each of the orders 14 and 15. In both cases they correspond to the families of a symmetric but sporadic Costas array, and actually the one at order 15 can be obtained from the one at order 14 by the addition of a corner dot. They are shown in Table 3. • Finally, when S = {(1, −1), (1, 1), (1, −2), (1, 2), (2, −1), (2, 1), (2, −2), (2, 2)}, all Costas arrays of order at most 27 register at least one distance vector in S. We can formulate then the following conjecture: Conjecture 2. The set mentioned in Problem 3 is S = {(1, −1), (1, 1), (1, −2), (1, 2), (2, −1), (2, 1), (2, −2), (2, 2)}. An exhaustive search for the set S mentioned in Problem 2, similar to the search for Problem 1, on all arrays of order n ∈ [27], shows that S = {(x, y) ∈ Z2 : x ∈ [3], y ∈ −[3] ∪ [3], (x, y) 6= (3, 3), (3, −3)}. Hence, the appropriate conjecture is: Conjecture 3. The set mentioned in Problem 2 is S = {(x, y) ∈ Z2 : x ∈ [3], y ∈ −[3] ∪ [3], (x, y) 6= (3, 3), (3, −3)}. 4. Common vectors between Golomb and Welch Costas arrays In the previous section we exhaustively searched the database of known Costas arrays, in those orders for which all Costas arrays are known today, for common vectors between pairs and formulated conjectures. In this section we will attempt to formulate and prove rigorously similar results for Golomb and Welch Costas arrays. Advances in Mathematics of Communications

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Order # NAKCAs # symmetric NAKCAs # NAKCAs by T4 1 1 1 1 2 0 0 0 3 0 0 0 4 0 0 0 5 4 4 4 6 4 4 0 7 4 4 4 8 12 4 0 9 16 0 0 10 44 4 0 11 60 4 0 12 272 0 0 13 416 16 0 14 524 20 0 15 428 20 4 16 584 16 0 17 432 16 0 18 256 8 0 19 224 8 0 20 160 0 0 21 96 0 0 22 40 0 0 23 32 0 0 24 0 0 0 25 0 0 0 26 0 0 0 27 4 4 4 Table 2. Number of NAKCAs per order. The columns of the arrays are, from left to right: order; total number of NAKCAs; number of symmetric NAKCAs (including all 4 members of each family, not only the two symmetric ones); number of NAKCAs produced by T4 . Order 14 15

Costas array 8 13 3 6 10 2 14 5 11 7 1 12 9 4 15 8 13 3 6 10 2 14 5 11 7 1 12 9 4

Table 3. Representatives of the two families of symmetric sporadic arrays at orders 14 and 15 that contain none of the vectors in {(1, −1), (1, 1), (1, −2), (1, 2), (2, −1), (2, 1)}; the representatives are clearly related through the addition of a corner dot. 4.1. Golomb Costas arrays. The following lemma is here of paramount importance: Lemma 1. In the notation of Theorem 2, the distance vector (u, v), |v| < q −2, 0 < u < q − 2 is not contained in the Golomb Costas array generated by the primitive roots a and b if au = bv in F(q). Advances in Mathematics of Communications

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Proof. We wish to solve the system of equations: ai + bj = 1, ai+u + bj+v = 1, by multiplying the first equation by au , then bv , and in each case subtracting them, we obtain the solution: au − 1 bv − 1 bj = u , ai = v . v a −b b − au When clearly au 6= 1 for the given range of values of u, so au = bv leads to no solution; otherwise, a unique solution exists. Remark 4. The existence of a unique solution in the lemma above when au 6= bv should not be taken to imply that the vector (u, v) is physically contained in the array: beware of the possibilities i < q − 1 ≤ i + u or j < q − 1 ≤ j + v, in which case the vectors are interrupted by a boundary of the aray: the Golomb equation really considers the array as a torus! Additional conditions will be needed to guarantee that the vector actually lies within the array. Theorem 3. Consider the Golomb array generated by the primitive roots a, b ∈ F(q); then 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

unless a = b, the vector (1, 1) lies in the array; unless a = b−1 , the vector (1, −1) lies in the array; unless b = a2 or a + b = 0, the vector (2, 1) lies in the array; unless b−1 = a2 or a + b−1 = 0, the vector (2, −1) lies in the array; unless a = b2 or a + b = 0, the vector (1, 2) lies in the array; unless a = b−2 or a + b−1 = 0, the vector (1, −2) lies in the array; unless a2 = b2 ⇔ a = ±b or a2 + b = 0 or a + b2 = 0, the vector (2, 2) lies in the array; unless a2 = b−2 ⇔ a = ±b−1 or a2 + b−1 = 0 or a + b−2 = 0, the vector (2, −2) lies in the array; unless a3 = b or a2 + a + b = 0 or a2 + ab + b = 0, the vector (3, 1) lies in the array; unless a3 = b−1 or a2 + a + b−1 = 0 or a2 + ab−1 + b−1 = 0, the vector (3, −1) lies in the array; unless b3 = a or b2 + a + b = 0 or b2 + ab + a = 0, the vector (1, 3) lies in the array; unless b−3 = a or b−2 + a + b−1 = 0 or b−2 + ab−1 + a = 0, the vector (1, −3) lies in the array.

Proof. The proof is inescapably quite repetitive in nature and consists of checking all possible cases: 1. If a 6= b, the only possibility for (1, 1) not lying within the array is either i = q − 2 = −1 or j = q − 2 = −1; the first equation in Lemma 1 in this case becomes: 1 a−1 = ⇔ a − b = ab − b ⇔ a = 0 or b = 1 b a−b which is impossible, while the second equation is the symmetric expression by swapping a and b, hence also impossible. 3. If b 6= a2 , (2, 1) will fail to be in the array only if it gets interrupted by the boundary; this would occur if i = q − 2 = −1 or i = q − 3 = −2 or Advances in Mathematics of Communications

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j = q − 2 = −1, leading to the equations:

1 b−1 1 b−1 1 a2 − 1 = , = , = . a b − a2 a2 b − a2 b a2 − b Among these, the second and the third lead to the trivial solutions b = 0 or a2 = 1, and a = 0 or b = 1, respectively, while the first yields the condition: b − a2 = ab − a ⇔ a2 + ab − a − b = 0 ⇔ (a − 1)(a + b) = 0 ⇔ a + b = 0

given that a 6= 1 as a primitive root. 7. If a2 6= b2 ⇔ a 6= ±b, (2, 2) will lie within the array unless i or j is equal to q − 2 = −1 or q − 3 = −2. The corresponding equations are:

1 b2 − 1 1 b2 − 1 1 a2 − 1 1 a2 − 1 = 2 , = 2 , = 2 , = 2 . 2 2 2 2 2 a b −a a b −a b a −b b a − b2 Among these, the second and the fourth lead to the trivial solutions b = 0 or a2 = 1, and a = 0 or b2 = 1, respectively, while the first gives:

b2 − a2 = b2 a − a ⇔ ab2 − b2 + a2 − a = 0 ⇔ (a − 1)(b2 + a) = 0 ⇔ b2 + a = 0

and the fourth the symmetric result in a and b, namely a2 + b = 0. 9. If a3 6= b, (3, 1) will lie within the array unless i = −1, −2, −3 or j = −1. The corresponding equations are: b−1 1 b−1 1 b−1 1 a3 − 1 1 = , = , = , = . a b − a3 a2 b − a3 a3 b − a3 b a3 − b Among these, the third and the fourth equation lead to the trivial solutions b = 0 or a3 = 1 and a = 0 or b = 1, respectively, while the first gives: b − a3 = ab − a ⇔ a3 + ab − a − b = 0 = (a − 1)(a2 + a + b) ⇔ a2 + a + b = 0,

as a 6= 1, and similarly the second gives: a3 + a2 b − a2 − b = 0 = (a − 1)(a2 + ab + b) ⇔ a2 + ab + b = 0. All remaining cases not shown follow from these by transposition or vertical flip. This completes the proof.

The large number of cases analyzed above allows us now to prove the following important theorem: Theorem 4. Any two sufficiently large Golomb Costas arrays have at least one of the following distance vectors in common: (1, 1), (1, −1), (2, 1), (2, −1), (3, 1), (3, −1), (3, 2), (3, −2). Proof. Consider two Golomb Costas arrays with a distance vector in common; if they both contain either (1, 1) or (1, −1) the proof is complete. Assume then that the first array, generated by some primitive roots a and b in some F(q), does not contain (1, 1), and that the second array, generated by some primitive roots α and β in some F(q ′ ), does not contain (1, −1). It follows, in accordance with Theorem 3, that a = b and that αβ = 1. Can both arrays contain the distance vector (2, 1)? The first would not if either a2 = a or 2a = 0; the former is impossible, as a is a primitive root, while the second is impossible unless q = 2m . The second, in turn, would not if either α−1 = α2 ⇔ α3 = 1 or α + α−1 = 0 ⇔ α2 = −1 ⇔ α4 = 1; the former excludes only the case q ′ = 3 and the latter only the case q ′ = 4, in which (2, 1) would not fit anyway. So, unless q = 2m , (2, 1) is a common vector. Advances in Mathematics of Communications

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Assume now that for the first array it holds true that a = b and q = 2m ; then, according to Theorem 3, this array cannot contain (1, 1), (2, 1), (1, 2), and (2, 2). Can both arrays contain (2, −1)? The second will unless α2 = α, which is ′ impossible, or 2α = 0, which is impossible unless q ′ = 2m . Assuming then that q ′ is not a power of 2, (2, −1) is a common vector. ′ Assuming, in addition, that q ′ = 2m , however, the second array cannot contain (1, −1), (2, −1), (1, −2), and (2, −2). Hence, we need to investigate the possibility that (3, 1) be a common distance vector. The first array will contain it iff none of the following equations hold: a3 = a, a3 + a2 = 0, and a2 + a = 0, which are clearly impossible as a is a primitive root. The second array will contain it iff none of the following equations hold: α4 = 1, α3 + 1 + α−1 + α = 0 ⇔ α4 + α2 + α + 1 = 0 ⇔ α3 = α2 + 1, α3 + α + α2 + α−1 = 0 ⇔ α4 + α3 + α2 + 1 = 0 ⇔ α3 = α + 1. The first of these equations excludes Golomb arrays of order 3, which are too small to contain (3, 1) anyway, while the other two describe two Golomb Costas arrays in F(8), one being the rotation be 180o of the other: their permutations can be readily found to be 245163 and 416235, and it can also be seen that they both contain the distance vector (3, −1). Can then (3, −1) be a common vector? It will, unless the first array satisfies the corresponding equations stated in Theorem 3. When is this the case? We could proceed as above, but we can use the result above as a shortcut: as the two arrays we found contain (3, 1) but not (3, −1), their vertical flips will contain (3, −1) but not (3, 1). Their corresponding permutations are then 532614 and 361542. We observe that all 4 of these arrays contain (3, 2) (as well as (3, −2)), and the proof is complete. 4.2. Welch Costas arrays. Theorem 5. All possible distance vectors are contained within a Welch Costas array (assuming the array wraps around at the boundaries). Proof. Consider the Welch Costas permutation generated by the primitive root g ∈ F(p), p prime, and the fixed constant c ∈ [p−1]−1: j = f (i) = g i−1+c mod p, i ∈ [p − 1]. If we want the distance vector (u, v) to appear in the array, we need to make sure the following system can be solved for i and j: j ≡ g i−1+c mod p, j + v ≡ g i+u−1+c mod p.

Let us first seek a solution to the very similar in appearance but very different system: j ≡ g i−1+c mod p, j + v ≡ g i+u−1+c mod p. Multiplying the first equation by g u and subtracting, we obtain: v v (1) j≡ u mod p, g i−1+c ≡ u mod p. g −1 g −1 Therefore, a unique solution exists for every distance vector. We now need to make sure that a particular vector actually lies within the array, so we formulate the counterpart of Theorem 3 for Welch arrays. Theorem 6. Consider the Welch Costas permutation generated by the primitive root g ∈ F(p), p prime, and the fixed constant c ∈ [p− 1]− 1: j = f (i) = g i−1+c mod p, i ∈ [p − 1]. Then: 1. unless g c − g c−1 ≡ 1 mod p, the vector (1, 1) lies in the array; Advances in Mathematics of Communications

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unless g c − g c−1 ≡ −1 mod p, the vector (1, −1) lies in the array; unless g c − g c−1 ≡ 2 mod p, the vector (1, 2) lies in the array; unless g c − g c−1 ≡ −2 mod p, the vector (1, −2) lies in the array; unless g c − g c−2 ≡ 1 mod p or g c+1 − g c−1 ≡ 1 mod p, (2, 1) lies in the array; unless g c − g c−2 ≡ −1 mod p or g c+1 − g c−1 ≡ −1 mod p, (2, −1) lies in the array; 7. unless g c − g c−2 ≡ 2 mod p, g c+1 − g c−1 ≡ 2 mod p, or g 4 ≡ 1 mod p, (2, 2) lies in the array; 8. unless g c − g c−2 ≡ −2 mod p, g c+1 − g c−1 ≡ −2 mod p, or g 4 ≡ 1 mod p, (2, −2) lies in the array.

2. 3. 4. 5. 6.

Proof. 1. This vector will fail to lie within the array iff its starting point is on the bottom or the right border of the array, namely iff i = p − 1 or j = p − 1. Setting 1 1 (u, v) = (1, 1) in (1), we obtain j ≡ mod p, g i−1+c ≡ mod p. g−1 g−1 Setting j = p − 1, we obtain g ≡ 0 mod p, which is impossible, while setting i = p − 1 we obtain g c − g c−1 ≡ 1 mod p. 2. This vector will fail to lie within the array iff its starting point is on the top or the right border of the array, namely iff i = p − 1 or j = 1. Setting 1 1 (u, v) = (1, −1) in (1), we obtain j ≡ − mod p, g i−1+c ≡ − mod p. g−1 g−1 Setting j = 1, we obtain g ≡ 0 mod p, which is impossible, while setting i = p − 1 we obtain g c − g c−1 ≡ −1 mod p. 3. This vector will fail to lie within the array iff i = p − 1, j = p − 2, or 2 j = p − 1. Setting (u, v) = (1, 2) in (1), we obtain j ≡ mod p, g i−1+c ≡ g−1 2 mod p. Setting j = p − 2, we obtain g ≡ 0 mod p, which is impossible; g−1 setting j = p− 1, we obtain g ≡ −1 mod p, which is impossible; finally, setting i = p − 1 we obtain g c − g c−1 ≡ 2 mod p. 5. This vector will fail to lie within the array iff its starting point (i, j) satisfies 1 i = p − 1, i = p − 2, or j = p − 1. The first equation leads to 2 ≡ g −1 1 g c−1 mod p ⇔ g c+1 − g c−1 ≡ 1 mod p; the second to 2 ≡ g c−2 mod p ⇔ g −1 1 g c − g c−2 ≡ 1 mod p; and the third to −1 ≡ 2 mod p ⇔ g ≡ 0 mod p g −1 which is impossible. 7. This vector will fail to lie within the array iff either i or j is equal to p − 1 or p − 2. These conditions lead to the 4 equations: 2 2 mod p, −2 ≡ 2 mod p, g2 − 1 g −1 2 2 ≡ 2 mod p, g c−2 ≡ 2 mod p. g −1 g −1

−1≡ g c−1

The first leads to g 2 ≡ −1 mod p ⇒ g 4 ≡ 1 mod p; the second to g ≡ 0 mod p which is impossible; the third to g c+1 − g c−1 ≡ 2 mod p; and the fourth to g c − g c−2 ≡ 2 mod p.

Cases 4, 6, and 8 follow directly from cases 3, 5, and 7, respectively. Advances in Mathematics of Communications

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As in the Golomb case, the large number of cases analyzed above allows us now to prove the following important theorem: Theorem 7. Any two sufficiently large Welch Costas arrays, as defined in Theorem 1, have at least one of the following distance vectors in common: (1, 1), (1, −1), (1, 2), (1, −2), (2, 1), (2, −1), (2, 2). Proof. Consider two Welch Costas arrays with a distance vector in common; if they both contain either (1, 1) or (1, −1) the proof is complete. Assume then that the first array, generated by some primitive root g ∈ F(p), does not contain (1, 1), and that the second array, generated by some primitive root h ∈ F(q), does not contain (1, −1). Note that it is impossible for a Welch Costas array to contain neither of (1, 1) and (2, 1), or neither of (1, −1) and (2, −1). Let us focus on the first case, as the second follows from a verbatim repetition of the argument for the first: if it were possible for neither of the vectors to be present, the array would need to satisfy, according to Theorem 6, both g c − g c−1 ≡ 1 mod p and one of g c − g c−2 ≡ 1 mod p, g c+1 − g c−1 ≡ 1 mod p. Either choice, however, leads to the trivial solution g = 0 or g = 1, which is impossible. We are actually assuming then that the first array contains neither (1, 1) nor (2, −1), while the second array contains neither (1, −1) nor (2, 1). Would it be possible for both not to contain (2, 2)? Once more, let us focus on the first array, as the equations for the second are almost the same. If the first array contains none of (1, 1), (2, −1), and (2, 2), it should satisfy g c − g c−1 ≡ 1 mod p c

one of g − g

c−2

c

≡ −1 mod p

and one of g − g 4

(1),

c−2

(a),

≡ 2 mod p (A),

g ≡ 1 mod p

(C).

g c+1 − g c−1 ≡ −1 mod p g

c+1

−g

c−1

≡ 2 mod p

(b), (B),

There are then 6 possible scenarios in total: 1aA: Impossible, unless −1 ≡ 2 mod p, which implies p = 3. 1aB: Summing (1) and (a) we get 2g 2 −g−1 ≡ 0 mod p ⇒ g ≡ −2−1 mod p, g ≡ 1 mod p, while (a) and (B) give g ≡ −2 mod p; this is impossible unless 2−1 ≡ 2 mod p ⇒ p = 3. 1aC: Summing (1) and (a) we get 2g 2 −g−1 ≡ 0 mod p ⇒ g ≡ −2−1 mod p, g ≡ 1 mod p, the latter being impossible. (C) implies p = 5, in which case g ≡ −2−1 ≡ 2 mod 5, and indeed g 4 ≡ 1 mod 5. Now, (1) yields 2c−1 ≡ 1 mod 5, namely c = 1. To sum up, the equations are compatible and specify the array 2431. 1bA: Summing (1) and (b) we get g 2 + g − 2 ≡ 0 mod p ⇒ g ≡ −2 mod p, g ≡ 1 mod p, while (b) and (A) give g ≡ −2−1 mod p; this is impossible unless 2−1 ≡ 2 mod p ⇒ p = 3. 1bB: Impossible, unless −1 ≡ 2 mod p, which implies p = 3. 1bC: Summing (1) and (b) we get g 2 + g − 2 ≡ 0 mod p ⇒ g ≡ −2 mod p, g ≡ 1 mod p, the latter being impossible. (C) implies p = 5, in which case g ≡ 1 ≡ 3 mod 5, 3 mod 5, and indeed g 4 ≡ 1 mod 5. Now, (1) yields 3c−1 ≡ 2 namely c = 2. To sum up, the equations are compatible and specify the array 4213, which is the rotation by 180o of 2431. Advances in Mathematics of Communications

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For the second array, we similarly end up with the two arrays 3124 and 1342, the vertical flips of the two arrays found above, which are also the rotation by 180o of each other. So, all pairs of Welch Costas arrays with a vector in common, except for 4 pairs of Welch arrays of order 4, have a common vector in (1, 1), (1, −1), (2, 1), (2, −1), (2, 2); and in those 4 pairs, we observe that (1, 2) and (1, −2) are always common vectors. We are not done yet with Welch arrays, however: the transposition of a Welch array cannot be generated by the general equation in Theorem 1, in contrast to the case of Golomb arrays. If we expand the family of Welch arrays to include transpositions as well, we need to check for common vectors pairs of transposed arrays as well as mixed pairs. Theorem 8. Any pair of Costas arrays consisting of a Welch array and a transposed Welch array with a common vector have a common vector in (1, 1), (1, −1), (2, 1). Proof. Assume the first array corresponds to the primitive root g ∈ F(p) and the parameter c, and that the second is the transposition of the array corresponding to the primitive root h ∈ F(q) and the parameter d. Assume further that the first array does not contain (1, 1), while the second does not contain (1, −1); according to Theorem 6, the first must then contain (2, 1), unless it is of order 2. Is it possible for the second not to contain (1, 2)? In such a case the second array should satisfy the equations hd − hd−1 ≡ −1 mod p and hd − hd−1 ≡ 2 mod p, implying that p = 3. Obviously, the common vectors between transposed Welch Costas arrays must lie within the transposed set of vectors of Theorem 7. Putting everything together, we obtain: Theorem 9. Any two Welch Costas arrays with a common distance vector have one of the following vectors in common: (1, 1), (1, −1), (1, 2), (2, 1), (2, −1), (1, −2), (2, 2), (2, −2). 4.3. Mixed pairs. What can we say about common vectors between a Golomb Costas array and a Welch Costas array? With all the results we have already obtained, the answer is quite simple to obtain: Theorem 10. A Welch and a Golomb Costas array with a vector in common always have a common vector among: {(1, 1), (1, −1), (1, 2), (1, −2), (2, 1), (2, −1)}. Proof. Assuming (1, 1) or (1, −1) is a common vector, the proof is complete. Hence, let us assume that the Welch array, corresponding to the primitive root g ∈ F(p) and the parameter c, does not contain (1, 1), while the Golomb array, corresponding to the primitive roots a, b ∈ F(q), does not contain (1, −1). It follows by Theorems 3 and 6 that (2, 1) is contained in the Welch array, and that a = b−1 . If the Golomb array does not contain (2, 1), either b = a2 or a + b = 0; it follows that either a3 = 1 or a2 = −1 ⇒ a4 = 1, implying that the Golomb array must be of order 2 or 3. There are only two Costas arrays that do not contain (1, 1), namely 312 and 231, and they both contain (2, −1) and (1, −2); now, Theorem 6 shows that any Welch array not containing (1, 1) contains (1, −2), and this completes this case. If we assume the symmetric scenario, namely that the Welch array does not contain (1, −1) and the Golomb array does not contain (1, 1), flipping both arrays Advances in Mathematics of Communications

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we are back in the previous case; therefore, the common vectors here are the previous ones with the sign of the second coordinate changed. This completes the proof. 4.4. A general theorem. Combining all of the results in this section, we get a general result on common vectors between Golomb and Welch arrays: Theorem 11. The smallest set of common vectors, closed under flips and transposition, with the property that any pair of sufficiently large Costas arrays, chosen among Golomb and Welch arrays only, has a vector in this set, is: {(1, 1), (1, −1), (1, 2), (2, 1), (2, −1), (1, −2), (2, 2), (2, −2), (3, 1), (3, −1), (1, 3), (1, −3), (2, 3), (2, −3), (3, 2), (3, −2)}. This the same set as in Conjecture 3. 5. NAKCAs Which types of Costas arrays should we seek NAKCAs in? The following results tell us that • T4 always yields symmetric NAKCAs, although not all symmetric NAKCAs are generated by T4 ; • all non-symmetric NAKCAs are necessarily sporadic, as no other systematic method yields NAKCAs. 5.1. Systematic construction of NAKCAs. Theorem 12. W1 and its derivative methods do not yield NAKCAs, except for orders less than 3. Proof. Consider the Welch array generated by the primitive root g ∈ F(p) and the parameter c; Theorem 6 reveals that either (1, 1) or (1, −1) (or both) must be contained in the array, unless −1 ≡ 1 mod p ⇒ p = 2. Our only hope then for a NAKCA Welch array is to remove the offending vector(s) by removing corner dots. Let us see the possibilities in detail: 1. Assume (1, −1) does not appear in the array: from Theorem 6, g c − g c−1 ≡ −1 mod p. We can remove (1, 1) iff it appears at a corner, and the only possibilities are either the top or the bottom left corner, where the starting dot will be (i, j) = (1, 1) or (i, j) = (1, p − 2). It follows from (1) that we need 1 to solve either 1 ≡ mod p, 1 ≡ g c mod p, whose solution is found g−1 1 immediately to be g = 2, c = 0, or −2 ≡ mod p, −2 ≡ g c mod p, g−1 p−1 p−3 whence g ≡ 2−1 mod p, and g c ≡ 2−c ≡ (2−1 )( 2 −1) ⇒ c = . In the 2 c c−1 −1 −1 former case, we obtain g − g ≡ 1 − 2 = 2 6≡ −1 mod p, unless p = 3, p−3 p−5 p+1 p+3 while in the latter case g c − g c−1 ≡ (2−1 ) 2 − (2−1 ) 2 ≡ 2 2 − 2 2 ≡ −2 − (−4) ≡ 2 6≡ −1 mod p, unless p = 3 as well. 2. Assume (1, 1) does not appear in the array; we can remove (1, −1) iff it appears at a corner, and the only possibilities are either the top or the bottom left corner, where the starting dot will be (i, j) = (1, 2) or (i, j) = (1, p − 1). If such an array existed, however, by flipping it vertically we would obtain an array of the case above, which we demonstrated it cannot exist. 3. Assume that both (1, 1) and (1, −1) appear, but occupy corner positions so they can both be removed. This means that: Advances in Mathematics of Communications

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(a) The Welch permutation begins with 1243: the first two entries imply that g = 2, c = 0, so that j ≡ 2i−1 mod p; the third entry is compatible with that. The fourth entry yields: 3 = 23 mod p ⇒ p = 5. It follows that a Welch permutation beginning with this sequence cannot be part of a longer permutation. The two vectors can be removed but then nothing remains! (b) The Welch permutation begins with 2134: then, the two first entries imply that g = 2−1 , c = −1 ≡ p − 2 mod (p − 1), so that j ≡ 22−i mod p; the third entry yields then 3 ≡ 2−1 mod p ⇒ p = 5 again. (c) The Welch permutation begins with 12 and ends with p − 1, p − 2: the beginning implies that j ≡ 2i−1 mod p, as we saw above, whence −2 ≡ 2p−2 mod p ⇔ −4 ≡ 1 mod p ⇒ p = 5 again. This completes the proof. Theorem 13. G2 and its derived methods do not yield NAKCAs when q > 4, except in the case of T4 , which yields symmetric NAKCAs. Proof. According to Theorem 3, at least one of (1, 1), (1, −1) must lie within the array. Hence, a G2 cannot be a NAKCA; we can only hope to obtain a NAKCA through the removal of corner dots. Let us consider then the Golomb array generated by the primitive roots a, b ∈ F(q), and let us see the possibilities in detail: 1. Assume the array does not contain (1, 1), so that a = b according to Theorem 3. Can (1, −1) be at a corner and thus be removed? Without loss of generality, we can check only the top and bottom left corners (rotate the array by 180o if necessary). So, the start of the distance vector would have to be at either (i, j) = (1, 2) or (i, j) = (1, q − 2). Following Lemma 1, either a−1 b−1 − 1 a−1 b−1 − 1 −1 b2 = , a = , or b = , a = . Given that a − b−1 b−1 − a a − b−1 b−1 − a 2 a = b as well, both equations of the former system become a + a = 1, while the latter system yields a2 + a = 1, a2 = a + 1, whence 2a = 0, which can only be true when q is a power of 2, and even then a2 + a = 1 ⇔ a2 = a + 1, so it is equivalent to the solution of the former system. 2. Assume the array does not contain (1, −1); a vertical flip would result to an array not containing (1, 1), thus bringing us to the previous case. Therefore, the solutions here are given by ab = 1, a2 + a = 1. This completes the proof. Note that, as a result of Theorems 12 and 13, NAKCAs not produced by Lempel arrays can clearly only be sporadic; in particular, all non-symmetric NAKCAs are sporadic. It would now be useful to identify those finite fields in which a primitive root a that satisfies a2 + a = 1 can occur. Here is a simple necessary condition: Theorem 14. The finite field F(p), p prime, can have a primitive root a such that a2 + a = 1 only if p mod 10 = ±1. Proof. The discriminant of the √ equation is clearly 5, and the solutions are given by −1 ± 5 the usual formula a = . Now, 2 is always invertible, but we still need 5 to 2 be a quadratic residue modulo p. This happens whenever p mod 10 = ±1. Notice, however, that, even in the case of the theorem, it may not be true that the solution a obtained is a primitive root. Note also that in F(q), where q = p2m , Advances in Mathematics of Communications

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√ p prime, m ∈ N∗ , 5 always exists, although once more there is no a priori reason for the a so computed to be a primitive root. 6. An extension to other vectors The methods we applied above to investigate whether particular distance vectors, such as (1, 1) and (1, −1), appear in a given (Welch or Golomb) Costas array can also be applied in a rather simple way to the more general case of the distance vectors (r, r) or (r, −r), 1 ≤ r ≤ n − 1, where n is the order of the array, if we are willing to forfeit the hard part of the investigation, namely finding whether the distance vector is actually contained within the array as opposed to “wrapping” around it. In this case, we already know the answer for Welch arrays, thanks to (1), while the theorem that follows treats the case of Golomb arrays: Theorem 15. Let A be a Golomb array of order q − 2 generated by the primitive roots a, b ∈ F(q), q being a power of a prime; then, A certainly contains the distance vector (r, r), 1 ≤ r ≤ q − 3, whenever the gcd (r, p − 1) = 1, and possibly for other values of r as well, unless a = b. In the particular case when ab = 1, A contains q−1 (r, r) for all values of r except when q odd. Further, A certainly contains the 2 distance vector (r, −r) whenever the gcd (r, p − 1) = 1, and possibly for other values of r as well, unless ab = 1. In the particular case when a = b, A contains (r, −r) q−1 for all values of r except when q odd. 2 Proof. The second half of the theorem can be obtained from the first by flipping the array horizontally (or vertically); therefore, we concentrate on the first half. Lemma 1 yields: br − 1 ar − 1 ai = r , bj = r . r b −a a − br Clearly, there is no solution iff ar = br (ar = 1, given the range of r and that a is a primitive root, is impossible). When is this the case? As both a and b are primitive roots, ∃s ∈ [q − 2], (s, q − 1) = 1 so than b = as , in which case ar = br ⇔ ar(s−1) = 1 ⇔ q − 1|r(s − 1). We distinguish the following cases: • s = 1: In this case a = b, and it leads to q − 1|0, which is true for all r; hence no vector of the form (r, r) is contained in A. • s > 1, (r, q − 1) = 1: In this case we need q − 1|s − 1 which is impossible given the range of s; hence, for these values of r, (r, r) is always contained in A. • s > 1, (r, q − 1) > 1: In this case it may be possible that q − 1|r(s − 1), and, for these values of r, (r, r) is not contained in A.

Finally, in the particular case when s = q − 2 ≡ −1 mod q − 1, implying that ab = 1, we can say even more: it follows that bj =

ar +1

ar

q−1 when q odd. 2 Hence, the vector (r, r) is contained in A, except for this particular value of r. When q is even, the field is of characteristic 2, so −1 = 1 and this exception does not occur.

which always has a unique solution unless ar = −1 ⇔ r =

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Remark 5. For each vector that does wrap around, there is exactly one other vector between the same dots that does not; further, both of these vectors are of the same type ((r, r) or (r, −r)). To conclude, with the possible exception of the missing vector in the case of Golomb arrays, exactly half of the vectors with equal or opposite coordinates are actually contained in a Welch or Golomb array. 6.1. An application: Non-attacking queens. Consider an extended n × n chessboard, n ∈ N∗ ; what is the maximal number of queens that can be placed on it so that no queen attacks another? We assume here that the chessboard is populated exclusively by queens, so that no other pieces, such as pawns, bishops etc. are present. This is a well known puzzle in mathematics, whose solution is also well known and dates back at least to Victorian Britain [6]: the maximum is always n queens, unless n = 2 or 3, where the answer is n − 1. Let us now focus on the case where n > 3: the solution clearly places n dots (queens) in a n × n rectangle, and consequently in such a way that each row and column contains exactly 1 dot (or else there would be two queens attacking one another); in other words, the solutions are permutation arrays. The number of these solutions as a function of n has also been studied [6], and seems to be rapidly increasing. The question we wish to consider here is the following: among all these solutions, is there one that also satisfies the Costas property? In other words, are there any Non-Attacking Queens Costas Arrays (NAQCAs)? This is a very old question indeed: S. Golomb considered it in 1982, and concluded that no configuration of non-attacking queens of order n ≤ 10 has the Costas property [9]. We tested all known Costas arrays of order n ≤ 27; no NAQCAs were found; note that a NAQCA is necessarily a NACKA, hence it suffices to check only the latter instead of rechecking all Costas arrays from scratch. Is the non-attacking queens property incompatible with the Costas property? So we conjecture: Conjecture 4. NAQCAs do not exist for n > 1. What we can prove, however, is somewhat weaker: Theorem 16. A NAQCA for n > 3 can only be sporadic. Proof. In order to avoid yet another tedious case by case study of the various possibilities, we offer a sketch of the argument. The reader will have no difficulty filling in the details. As a first step, note that a Costas array contains a pair of attacking queens iff it contains a distance vector of either the form (r, r) or (r, −r), r < n. As a second step, observe that Theorem 15, (1), and Remark 5 together imply that a) Golomb arrays generated in F(q) contain all vectors of the form (r, r) and (r, −r), where (r, q − 1) = 1, except when ab = 1 or a = b, where one of the families is missing, and that b) Welch arrays generated in F(p) contain exactly half of the totality of the vectors (r, r) and (r, −r), where 1 ≤ r ≤ p − 2. To sum up, both families contain “plenty” of vectors that allow pairs of queens to attack each other. As a third step, observe that the various derived Golomb and Welch methods that create new arrays through removal of corner (combinations of) dots still contain plenty of the above vectors, as long as the order is sufficiently large (smaller orders can –and have, as we mentioned above– been checked exhaustively, yielding no NAQCAs). As for the arrays created through addition of corner dots, they obviously contain at least as many offending pairs as the original array. To sum up, any Costas array created by any of the Golomb or Welch methods contains pairs of attacking queens. Hence, a NAQCA, if it exists at all, can only Advances in Mathematics of Communications

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be a sporadic Costas array (as long as it is sufficiently large; order 1 is an obvious exception). 7. Conclusion and future direction We investigated the distance vectors in individual Costas arrays (of various categories), and in pairs of Costas arrays. We offered general results by simulation, and rigorous results for algebraically constructed arrays. We were especially interested in pairs of Costas arrays with no distance vector in common (“orthogonal” pairs of Costas arrays), and the evidence we discovered seems to suggest that such pairs do not exist if the order of the arrays is large enough (5 or higher). Subsequently, we were led to study NAKCAs, as a subproblem. We saw that most NAKCAs are sporadic, with the exception of those (symmetric) Costas arrays of the Golomb-Lempel family constructed by the relatively rarely applicable submethod T4 . As a final step, we attempted to link the Costas property with the non-attacking queens problem, and we studied the NAQCAs, a very elusive sub-species of Costas arrays indeed, as only one (trivial) member was discovered! We were able to rule that any nontrivial NAQCA, if one exists, must be sporadic, but we would be willing to bet that such arrays do not exist at all (for finite orders). Our results constitute little progress towards the general problem itself of the distance vectors present within a Costas array. The (tedious and repetitive) methods used above can be extended to the derived algebraic constructions (such as W2 or T4 ), but the biggest problem we will still be facing is the classification of the distance vectors of the sporadic arrays (which, by the way, is also necessary for a definitive ruling on the existence of nontrivial NAQCAs, in view of Theorem 16): this task clearly requires some understanding of the way in which the constraining equations of the Costas property shape the distance vectors, and we feel that such an understanding is currently eluding us. Acknowledgements The authors would like to thank Prof. Nigel Boston for bringing reference [9] to their attention in relation with the existence of NAQCAs. This material is based upon works supported by the Science Foundation Ireland under Grant No. 05/YI2/I677, 06/MI/006 (Claude Shannon Institute), and 08/RFP/MTH1164.

References [1] J. P. Costas, Medium constraints on sonar design and performance, Technical Report Class 1 Rep. R65EMH33, GE Co., 1965. [2] J. P. Costas, A study of detection waveforms having nearly ideal range-doppler ambiguity properties, Proceedings of the IEEE, 72 (1984), 996–1009. [3] K. Drakakis, A review of Costas arrays, J. Appl. Math., 2006 (2006), 32. [4] K. Drakakis, On some properties of Costas arrays generated via finite fields, IEEE CISS, 2006. [5] K. Drakakis, R. Gow and S. Rickard, Interlaced Costas arrays do not exist, Mathematical Problems in Engineering, 2008 (2008). [6] H. E. Dudeney, The eight queens, §300 in “Amusements in Mathematics,” Dover, New York, (1970), 89 and 95–96. [7] A. Freedman and N. Levanon, Any two N × N Costas signals must have at least one common ambiguity sidelobe if N > 3 — A proof, Proceedings of the IEEE, 73 (1985), 1530–1531. Advances in Mathematics of Communications

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[8] S. W. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13–21. [9] S. W. Golomb and H. Taylor, Two-dimensional synchronization patterns for minimum ambiguity, IEEE Trans. Inform. Theory, 28 (1982), 600–604.

Received September 2008; revised January 2009. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]

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