Cross-Correlation Properties of Costas Arrays and Their Images under Horizontal and Vertical Flips

Abstract

We consider the cross-correlation of a Costas array with its image under a horizontal and/or a vertical flip. We propose and prove several bounds on the maximal cross-correlation and on its value at the origin, for both general Costas arrays and for algebraically constructed ones.

1. Introduction

Costas arrays were introduced as frequency hopping patterns with ideal autoambiguity properties by Dr. Costas [1, 2], in an effort to improve SONAR performance. Unable to find a general construction technique himself, he approached Professor Solomon Golomb, who published [3] and proved [4] two generation techniques for Costas arrays, both based on the theory of finite fields, known as the Welch and the Golomb-Lempel methods, respectively. These are still the only general construction methods for Costas arrays available today.

In multiuser and multiplexing systems it is desirable that the signals used have low cross-correlation, in order to minimize cross-talk. While Costas arrays are known for (and, indeed, defined by) their ideal autocorrelation, they typically exhibit poor cross-correlation. Subfamilies of Costas arrays with low cross-correlation are, therefore, particularly suitable for such applications. In this paper, we examine the cross-correlation of pairs of Costas arrays related by a horizontal and/or a vertical flip, in order to identify array pairs with good cross-correlation properties. Such a small family will be sufficient in some applications, for example, when a small number of SONARs/RADARs (4 or less) operate in the same geographical area (otherwise larger families will need to be considered, but this study lies outside the scope of this paper).

Regarding the layout of the paper, in Section 2 we give the definition of Costas arrays and describe the construction methods available. In Section 3, we review the literature on cross-correlation properties of Costas arrays. In Section 4, we prove the reflective symmetry property of Welch Costas arrays. Section 5 states and proves bounds on the cross-correlation of pairs of Costas arrays in the same family.

2. Costas Arrays: Definition and Algebraic Constructions

Definition 2.1. Let [𝑛]∶={1,…,𝑛},𝑛∈ℕ and consider a bijection 𝑓∶[𝑛]→[𝑛];𝑓 is a Costas permutation if and only if for all 𝑖,𝑗,𝑘 such that 1≤𝑖,𝑗,𝑖+𝑘,𝑗+𝑘≤𝑛,𝑓(𝑖+𝑘)−𝑓(𝑖)=𝑓(𝑗+𝑘)−𝑓(𝑗)⇒𝑖=𝑗or𝑘=0. (2.1)A Costas array 𝐴^𝑓 is the permutation array corresponding to a Costas permutation 𝑓. This correspondence is a matter of convention: we choose here the convention that the 𝑗th element of the permutation indexes the position of the (unique) 1 in the 𝑗th column of the array, 𝑗∈[𝑛], counting from top to bottom in the usual array convention: 𝑓(𝑖)=𝑗⇔𝑎^𝑓_𝑗,𝑖=1. It is customary to refer to (and denote) the 1s of a permutation array as “dots” and to 0s as “blanks.” Definition 2.1 is equivalent to the statement that all distance vectors (𝑓(𝑖)−𝑓(𝑗),𝑖−𝑗), where 𝑖>𝑗, between pairs of dots in a Costas array are distinct (note that we take the second coordinate to be nonnegative).

An example Costas array is shown in the top left of Figure 1. Note that the image of a Costas array under rotation by 90^∘, horizontal or vertical flips, and transposition is still a Costas array: each Costas array generates a family of eight members (or four, if the array is symmetric) also shown in the figure. The notation used signifies the generation of this family using two operators: 𝑅, a 90^∘ rotation counterclockwise (CCW), and 𝑇, a vertical flip. We also define for convenience 𝑆=𝑅²𝑇 to be the horizontal flip, in which case we also have that 𝑅²=𝑆𝑇=𝑇𝑆, the double (horizontal and vertical) flip. The numbers under each array correspond to the permutation indexing it.

(a)

(b)

Figure 1 

Labeling of rotations and reflections of a Costas array. 𝐼, the original array; 𝑅, a 90∘ rotation CCW; 𝑆𝑇=𝑅2, a 180∘ rotation; 𝑅3, a 270∘ rotation CCW; 𝑅𝑇, a reflection about the main antidiagonal; 𝑆=𝑅2𝑇, a horizontal flip; 𝑅3𝑇, a reflection about the main diagonal; and 𝑇, a vertical flip.

Every Costas array of size 𝑛≤27 has been found by exhaustive computer search [5–7]: they add up to about 150 000 arrays in total.

The two known construction methods for Costas arrays are based on finite field techniques; each one has several submethods/variants [3, 4, 8]. In the sequel we will work exclusively with the main methods.Theorem 2.2 (Welch construction 𝑊₁(𝑝,𝛼,𝑐)). Let 𝑝 be a prime, let 𝛼 be a primitive root of the finite field 𝔽(𝑝) of 𝑝 elements, and let 𝑐∈[𝑝−1]−1 be a constant; then, the function 𝑓∶[𝑝−1]→[𝑝−1] where 𝑓(𝑖)=𝛼^{𝑖−1+𝑐}mod𝑝 is a bijection with the Costas property.The reason for the presence of −1 in the exponent is that when 𝑐=0, 1 is a fixed point 𝑓(1)=1. We refer to arrays generated with 𝑐≠0 as circular shifts of the array generated by 𝑐=0 for the same 𝑝 and 𝛼. Welch Costas arrays have antireflective symmetry (see below, and see also [9]).Definition 2.3. Let 𝑓∶[2𝑛]→[2𝑛];𝑓 has antireflective symmetry if and only if 𝑓(𝑛+𝑖)+𝑓(𝑖)=2𝑛+1,𝑖∈[𝑛], (2.2)which means, in other words, that the right half of 𝐴^𝑓 is the vertical flip of the left half.Theorem 2.4 (Golomb construction 𝐺₂(𝑝,𝑚,𝑎,𝑏)). Let 𝑝 be a prime, 𝑚∈ℕ, and let 𝛼,𝛽 be primitive roots of the finite field 𝔽(𝑝^𝑚) of 𝑞=𝑝^𝑚 elements; then, the function 𝑓∶[𝑞−2]→[𝑞−2] where 𝛼^𝑖+𝛽^𝑓(𝑖)=1 is a bijection with the Costas property.

If 𝛼=𝛽, the array is known as a Lempel Costas array, and is symmetric about the main diagonal.

3. Cross-Correlations of Costas Arrays

Definition 3.1. Let 𝑓,𝑔∶[𝑛]→[𝑛],𝑛∈ℕ, and let 𝑢,𝑣∈ℤ; the cross-correlation between 𝑓 and 𝑔 at (𝑢,𝑣) is defined asΨ_𝑓,𝑔(𝑢,𝑣)=|{(𝑓(𝑖)+𝑢,𝑖+𝑣)∶𝑖∈[𝑛]}∩{(𝑔(𝑖),𝑖)∶𝑖∈[𝑛]}|. (3.1)In other words, in order to find the cross-correlation of two Costas arrays 𝐴^𝑓 and 𝐴^𝑔 of order 𝑛 at (𝑢,𝑣), we place the two arrays on top of each other, slide the first 𝑣 columns horizontally and 𝑢 rows vertically, and count the number of pairs of dots that overlap. We will also use the notation Ψ_𝐴^𝑓_,𝐴^𝑔 instead of Ψ_𝑓,𝑔.

There are few previous works on cross-correlations of Costas arrays in the literature. The closest to the subject of this paper is O'Carroll et al. [10] (further details about its relevance to the present work will be given in Section 5). Titlebaum and Maric [11] proved that the maximal cross-correlation of two Welch or Lempel Costas arrays generated from reciprocal primitive roots of an odd prime is 2. They also proved that the upper bound on the maximum of the cross-correlation of two Welch Costas arrays of order 𝑛 is 𝑛/2, and established that primes of certain forms had good or bad cross-correlation bounds, for example, they showed that the upper bound on the maximum of the cross-correlation of a pair of Welch Costas arrays is realized for primes equivalent to 1 modulo 4.

Drumheller and Titlebaum [12] derived upper bounds on the maximal cross-correlation of two Golomb or two Welch Costas arrays, both in terms of the relative powers of the generating primitive elements.

Rickard has previously proposed that the upper bound on the maximal cross-correlation of a pair of Golomb arrays of order 𝑛, taken over all such possible pairs of different arrays, is at most (𝑛−1)/2 [13]. He also proposed that (a) for primes 𝑝 of the form 2𝑟+1,𝑟 prime (known as safe or Germain primes), this quantity exhibits local minima, and that (b) the upper bound on the maximal cross-correlation of a pair of Golomb arrays of order 𝑝−2, taken over all such possible pairs of different arrays, is one less than the corresponding quantity for pairs of Welch arrays of order 𝑝−1, taken over all pairs of different arrays that are not circular shifts of each other, as long as 𝑝 is not a safe prime.

Finally, Etzion has demonstrated that the maximum of the cross-correlation of any Costas array with its image under rotation by 180^∘ is exactly 2 [14].

4. A Symmetry Property of the Welch Arrays

Welch Costas arrays, as mentioned above, have anti-reflective symmetry (see Definition 2.3), also known as glide-reflection symmetry (G-symmetry) [14]. For completeness, we present a proof of this property.

Theorem 4.1. A Welch Costas array has antireflective symmetry.Proof. Consider the Welch Costas permutation 𝑓 generated by 𝛼 in 𝔽(𝑝); then, for all 𝑖∈[(𝑝−1)/2],𝑓(𝑖)+𝑓(𝑖+𝑝−12)=𝛼^{+𝑖−1+𝑐}+𝛼^{(𝑝−1)/2+𝑖−1+𝑐}=𝛼^{+𝑖−1+𝑐}+𝑝−𝛼^{𝑖−1+𝑐}=𝑝, (4.1)in view of the fact that 𝛼^(𝑝−1)/2≡−1mod𝑝. This completes the proof.

We will make use of this property later in the paper. It should be noted that this symmetry does not characterize the Welch Costas arrays, that is, there exist non-Welch arrays with this property. Some conditions on the existence of arrays with antireflective symmetry were established in [14].

5. Proof of Some Correlation Bounds

In this section we present upper bounds on the cross-correlation values of Costas arrays related by a flip. More specifically, we divide our results in two groups: those dealing with the maximum of the cross-correlation and those dealing with the cross-correlation at the origin.

5.1. Bounds on Cross-Correlation Maximum

Freedman and Levanon [15] proved that, for 𝑛>3, the cross-correlation between any two arrays of the same size will have at least 2 hits. We present a special case where the cross-correlation between a pair of arrays is exactly 2: the cross-correlation between any Costas array and its 180^∘ rotation has a maximum value of 2.

Theorem 5.1. Let 𝐼 be a Costas array of order 𝑛>1; then max_(𝑢,𝑣)Ψ_𝐼,𝑅²(𝑢,𝑣)=2.Proof. The key observation is that a rotation by 180^∘ generates a Costas array whose distance vectors are the same as in the original array: letting two dots lie at 𝐴=(𝑓(𝑖),𝑖),𝐵=(𝑓(𝑗),𝑗), where 𝑖>𝑗, their images in the rotated array lie at 𝐴^′=(𝑛+1−𝑓(𝑖),𝑛+1−𝑖),𝐵^′=(𝑛+1−𝑓(𝑗),𝑛+1−𝑗),𝑛 being the order of the array. The distance vectors 𝐵𝐴 and 𝐴^′𝐵^′ are equal: (𝑓(𝑖),𝑖)−(𝑓(𝑗),𝑗)=(𝑓(𝑖)−𝑓(𝑗),𝑖−𝑗) and (𝑛+1−𝑓(𝑗),𝑛+1−𝑗)−(𝑛+1−𝑓(𝑖),𝑛+1−𝑖)=(𝑓(𝑖)−𝑓(𝑗),𝑖−𝑗). It follows that a shift will align these pairs, whence max_(𝑢,𝑣)Ψ_𝐼,𝑅²(𝑢,𝑣)≥2 (but note that the shift aligns 𝐴 with 𝐵^′ and 𝐵 with 𝐴^′).
Assume now that, for this given shift, a third pair of dots become aligned as well, say 𝐶 and 𝐾, where 𝐶 is a point in 𝐼 and 𝐾 a point in 𝑅²: then 𝐾 has a pre-image in 𝐼, say a point 𝑘. We claim that the points 𝐴,𝐵,𝐶, and 𝑘 form a parallelogram, violating the Costas property: indeed, by the triple alignment we get 𝐴^′𝐾=𝐵𝐶 and 𝐵^′𝐾=𝐴𝐶, while, by the observation above, 𝐴^′𝐾=𝑘𝐴 and 𝐵^′𝐾=𝑘𝐵. It follows that 𝐵𝐶=𝑘𝐴,𝐴𝐶=𝑘𝐵, hence 𝐴𝑘𝐵𝐶 is a parallelogram and the proof is complete.

As mentioned earlier, this result has also been derived by Etzion ([14, Theorem 2]); we chose to present it here for completeness, offering a more concise proof. O'Carroll et al. [10] sketched a proof that the maximal cross-correlation of any two amongst a Golomb array and its 𝑆,𝑇, and 𝑆𝑇 flips is identically 2.

We now prove two bounds for Welch Costas arrays of order 𝑛: first, we show that the peak of the cross-correlation of a Welch Costas array and its vertical flip 𝑇 is 𝑛/2. Then we consider the case of a Welch Costas array and its horizontal flip 𝑆 which yields a rather better result.

Theorem 5.2. For any antireflective Costas array 𝐼 of (even) order 𝑛 and its vertical flip 𝑇,max_(𝑢,𝑣)Ψ_𝐼,𝑇𝑛(𝑢,𝑣)=2. (5.1)In particular, the result holds true for any Welch Costas array generated in 𝔽(𝑝) (where 𝑛=𝑝−1).Proof. We prove this property by considering the form of three distinct regions of the cross-correlation. Let 𝑓 be the permutation corresponding to 𝑖. Then, we show that
(i)for all 𝑢,Ψ_𝐼,𝑇(𝑢,𝑣)≤1 when 𝑣≠𝑛/2,(ii)Ψ_𝐼,𝑇(𝑢,𝑛/2)=0 when 𝑢≠0,(iii)Ψ_𝐼,𝑇(0,𝑛/2)=𝑛/2. Fixing 𝑢 and 𝑣, we find the number of pairs (𝑖,𝑗) for which𝑛(𝑓(𝑖),𝑖)=(𝑢,𝑣)+(𝑛+1−𝑓(𝑗),𝑗)⇒𝑓(𝑣+𝑗)=𝑢+𝑛+1−𝑓(𝑗)=𝑢+𝑓(𝑗±2), (5.2)where “+” is chosen if 𝑗≤𝑛/2, and “−” otherwise.
Assuming 𝑣≠𝑛/2, this equation has at most one solution 𝑗 because of the Costas property. Assuming 𝑣=𝑛/2, the equation becomes 𝑢=0, which is satisfied for no 𝑗 if we choose 𝑢≠0, but is satisfied for all 𝑗∈[𝑛/2], that is 𝑛/2 values of 𝑗, if we choose 𝑢=0. This completes the proof.

Theorem 5.3. For any Welch Costas array 𝐼 and its horizontal flip 𝑆,max_(𝑢,𝑣)Ψ_𝐼,𝑆(𝑢,𝑣)=2. (5.3)Proof. Let 𝑓 be the Welch permutation generated in 𝔽(𝑝) by the primitive root 𝛼 and the parameter 𝑐. For fixed values of 𝑢 and 𝑣, we need to find the number of pairs (𝑖,𝑗) for which(𝛼^{𝑖−1+𝑐}mod𝑝,𝑖)=(𝑢,𝑣)+(𝛼^{𝑝−𝑗−1+𝑐}mod𝑝,𝑗) (5.4)whence we get the two equations𝛼^{𝑖−1+𝑐}mod𝑝=𝑢+𝛼^{𝑝−𝑗−1+𝑐}mod𝑝,𝑖=𝑣+𝑗, (5.5)namely𝛼^{𝑣+𝑗−1+𝑐}mod𝑝=𝑢+𝛼^{𝑝−𝑗−1+𝑐}mod𝑝. (5.6)If this equation is true, it is also true mod𝑝, whence𝛼^{𝑣+𝑗−1+𝑐}≡𝑢+𝛼^𝑐−𝑗mod𝑝⟺𝛼^{𝑣−1+𝑐}𝛼^2𝑗−𝑢𝑔^𝑗−𝛼^𝑐≡0mod𝑝. (5.7)Setting 𝛼^𝑗=𝑥, we obtain𝛼^{𝑣−1+𝑐}𝑥²−𝑢𝑥−𝛼^𝑐≡0mod𝑝. (5.8)This is an equation of the second degree in a field, and therefore it has at most 2 solutions for 𝑥 and thus for 𝑗. Note, however, that not all solutions of this equation need actually also be solutions of (5.6). Note also that, as both 𝐼 and 𝑆 are Costas arrays, Freedman and Levanon's result [15] guarantees that there are two solutions for at least one pair of values of 𝑢 and 𝑣.

Theorem 5.4. For any Costas array 𝐼 of order 𝑛>1,max_(𝑢,𝑣)Ψ_𝐼,𝐺𝑛(𝑢,𝑣)≤2or𝑛−12, (5.9)depending on whether 𝑛 is even or odd, respectively, where 𝐺 is either 𝑆 or 𝑇.Proof. Assume 𝑛 even, and assume there exists a shift for which a subset 𝐴 of (𝑛/2)+1 dots of 𝐼 can be aligned with a subset 𝑊 of equally many dots of 𝑇.𝐼 also contains a subset 𝐵 of (𝑛/2)+1 dots, namely the pre-image of 𝑊: the distance vectors in 𝐵 are the vertical flips of the distance vectors in 𝐴.
Observe now that, as no two dots of a Costas array lie in the same row or column, each distance vector is different from its vertical flip. Assume 𝐴 contains both a vector 𝑥 and its vertical flip 𝑥^′, where, without loss of generality, the starting point of 𝑥 lies to the left of the starting point of 𝑥^′. Then 𝐵 contains both vectors as well, but now the starting point of 𝑥 lies to the right of the starting point of 𝑥^′. This implies that the two pairs of equal vectors cannot both be collocated, hence 𝐼 contains at least two equal vectors and is not Costas, a contradiction.
It follows that distance vectors between points of 𝐴 are disjoint from distance vectors between points of 𝐵. But this is absurd, since, by the pigeonhole principle, 𝐴 and 𝐵 have at least 2 dots in common, hence they necessarily share a common distance vector.
Assuming 𝑛 odd, and considering a subset 𝐴 of (𝑛−1)/2+1 dots of 𝐼, we end up, as before, with another subset 𝐵 of (𝑛−1)/2+1 dots in 𝐼, so that each set contains the vertical flips of the distance vectors present in the other. But now the pigeonhole principle only guarantees the two sets have at least one dot in common, so it is necessary to count more carefully. Indeed, if we assume that the shift aligning 𝐴 and 𝐵 has a nonzero vertical component, say 𝑢≠0, it follows that |𝑢|≥1 dots can belong to neither 𝐴 nor 𝐵, whence |𝐴∩𝐵|≥2, that is, they share a common distance vector, which is impossible.
This implies that the shift aligning 𝐴 and 𝐵 is purely horizontal, so that 𝐴 and 𝐵 are vertical flips of each other. The only dot that could possibly lie in 𝐴∩𝐵 then is the (unique) dot of the central row, and since at least one dot must exist in 𝐴∩𝐵 (or else 𝐼 would have at least 𝑛+1 dots), we deduce that no shift is possible. Hence, (𝑛−1)/2+1=1⇔𝑛=1 is the only possibility.
The argument for 𝐺=𝑆 is completely analogous. This completes the proof.

Note that, for 𝑛 even, the upper bound is actually achieved for a Welch Costas array, while, for 𝑛 odd, the upper bound is again achieved for a Costas array resulting from the removal of the corner dot from a Welch Costas array generated with 𝑐=0, in both cases due to the antireflective symmetry.

5.2. Bounds on Origin Value of Cross-Correlation

We now consider some of the bounds regarding the values of the cross-correlation at the origin, that is, the number of hits for perfect overlay of two arrays. The reason for our particular interest in the value of the cross-correlation at the origin is, on the one hand, practical, as it becomes the only relevant value in a multiuser system where users' clocks are synchronized, and, on the other hand, theoretical, as it is often much simpler to compute or estimate than a general value, as we are about to see.

First, we show that a Costas array of even order and its vertical or horizontal flip (𝑆 and 𝑇, resp.) do not overlap, while the corresponding result for odd orders is that they overlap at one point only. Note that we already derived and used part of this result in the proof of Theorem 5.4.

Theorem 5.5. For any Costas array of order 𝑛,Ψ_𝐼,𝐵(0,0)=𝑛mod2, (5.10)when array 𝐵 is either 𝑆 or 𝑇.Proof. Since a Costas array contains one dot per column and row, Ψ_𝐼,𝑇(0,0) will equal the number of dots a vertical flip leaves fixed: if 𝑛 is even there is no fixed dot, whereas if 𝑛 is odd exactly one dot remains fixed, namely the one lying on the central row. The horizontal flip is analogously treated. This completes the proof.

We now sketch a proof that a symmetric Costas array of even order and its rotation by 90^∘ in either direction (𝑅 and 𝑅³) do not overlap, while the corresponding result for odd order is that they overlap at one point only.

Theorem 5.6. For any symmetric Costas array of order 𝑛,Ψ_𝐼,𝐵(0,0)=𝑛mod2 (5.11)when array 𝐵 is one of 𝑅 or 𝑅³.Proof. We can show this by observing that a symmetric array's rotation by ±90^∘ is equivalent to a horizontal or vertical flip and then invoking Theorem 5.5.

Next, we prove that if we overlay a Welch Costas array with its 𝑅² rotation, the number of hits depends on the array order.

Theorem 5.7. For any Welch Costas array 𝐼,Ψ_𝐼,𝑅²(0,0)=0,for𝑝≡1mod4,2,for𝑝≡3mod4. (5.12)Proof. A Welch Costas array of size 𝑛=𝑝−1 contains dots at positions(𝑓(𝑖),𝑖),𝑖∈[𝑝−1]where𝑓(𝑖)=𝛼^{𝑖−1+𝑐}mod𝑝 (5.13)for prime 𝑝 and a primitive element 𝛼 of 𝔽(𝑝). Its 𝑅² rotation contains dots at positions (𝑝−𝑓(𝑗),𝑝−𝑗),𝑗∈[𝑝−1].
The value at the origin of the cross-correlation is equal to the largest possible number of solutions of the simultaneous equations𝑓(𝑖)=𝑝−𝑓(𝑗),𝑖=𝑝−𝑗. (5.14)Substituting the latter into the former and using (5.13), we get, after some manipulation,⟺⟺𝛼^{𝑖−1+𝑐}mod𝑝+𝛼^{𝑝−𝑖−1+𝑐}mod𝑝=𝑝⟺𝛼^{𝑖−1+𝑐}+𝛼^−𝑖+𝑐≡0mod𝑝⟺𝛼^{𝑖−1+𝑐}≡𝛼^{−𝑖+(𝑝−1)/2+𝑐}mod𝑝⟺𝑖−1≡−𝑖+𝑝−123mod(𝑝−1)⟺either𝑖−1=−𝑖+2(𝑝−1)⟺𝑖=3𝑝−14or𝑖−1=−𝑖+𝑝−12⟺𝑖=𝑝+14. (5.15)As 𝑖 is an integer, these equations can be satisfied if and only if 𝑝≡3mod4. Therefore, when 𝑝 is of this form, we have two solutions, but when 𝑝≡1mod4, there are no solutions.

Theorem 5.8. For a Golomb Costas array 𝐼 of order 𝑛,Ψ_𝐼,𝑅²(0,0)=𝑛mod3. (5.16)Proof. A Golomb Costas array of order 𝑛=𝑞−2 has dots at positions (𝑓(𝑖),𝑖), where𝛼^𝑖+𝛽^𝑓(𝑖)=1,𝑖∈[𝑞−2] (5.17)for 𝑞=𝑝^𝑚,𝑝 prime, integer 𝑚, and primitive elements 𝛼 and 𝛽. Its 𝑅² rotation has dots at positions (𝑞−1−𝑓(𝑗),𝑞−1−𝑗),𝑗∈[𝑝−2].
The value of the cross-correlation at the origin is the number of solutions of the simultaneous equations 𝑓(𝑖)+𝑓(𝑗)=𝑞−1 and 𝑖+𝑗=𝑞−1. It follows that𝛼^𝑗+𝛽^𝑓(𝑗)=1,𝑓(𝑗)=𝑞−1−𝑓(𝑖),𝑗=𝑞−1−𝑖⇒𝛼^{𝑞−1−𝑖}+𝛽^{𝑞−1−𝑓(𝑖)}=1⟺𝛼^−𝑖+𝛽^{−𝑓(𝑖)}⟺1=1𝛼^𝑖+1𝛽^𝑓(𝑖)⟺𝛼=1_𝑖+𝛽^𝑓(𝑖)𝛼^𝑖𝛽^𝑓(𝑖)⟺1=1𝛼^𝑖𝛽^𝑓(𝑖)=1. (5.18)We therefore end up with two simultaneous equations:𝛼^𝑖𝛽^𝑓(𝑖)=1,𝛼^𝑖+𝛽^𝑓(𝑖)=1⟺(𝛼^𝑖)²−𝛼^𝑖+1=0𝛽^𝑓(𝑖)=1−𝛼^𝑖. (5.19)Setting 𝑥=𝛼^𝑖, we obtain the quadratic𝑥²−𝑥+1=0. (5.20)Now we consider three cases.
(i)𝑛≡1mod3⇔𝑞≡0mod3: 𝑞 is a power of 3, so (5.20) becomes𝑥²+2𝑥+1=0⟺(𝑥+1)²=0⟺𝑥=2. (5.21)Since only solution exists, Ψ_𝐼,𝑅²(0,0)=1.(ii)Otherwise, 𝑥 solves 𝑥²−𝑥+1=0 if and only if −𝑥 solves 𝑥²+𝑥+1=0⇔(𝑥³−1)/(𝑥−1)=0⇔𝑥³=1,𝑥≠1. This has solutions in 𝔽(𝑞) if and only if 3∣𝑞−1⇔𝑛≡2mod3: (a) 𝑛≡2mod3: (5.20) has 2 roots in 𝔽(𝑞). (b) 𝑛≡0mod3: (5.20) has 0 roots 𝔽(𝑞).

6. Conclusion

Families of Costas arrays with low cross-correlation are useful in a multiuser setting, where the necessity for ideal autoambiguity for optimal RADAR/SONAR performance is combined with the need to minimize cross-interference between different users. We focused on the special case of the cross-correlation of two Costas arrays related by a horizontal and/or a vertical flip and considered both general Costas arrays and more specific algebraically constructed subfamilies.

On many occasions the cross-correlation was found to be low (2 or less), implying that for a small number of users it is possible to use flipped variants of the same array without significant interference. Our options increase further if we tie all users to a common clock, in which case the only cross-correlation value that becomes relevant is that at the origin.

For a larger number of users, larger families of Costas arrays need to be considered; we relegate this for future work. We have also collected empirical results regarding the cross-correlation of a Costas array with its image under rotation by ±90^∘, or under transposition (and flips and/or rotations thereof), but so far we have been unable to find provable formulas for these results.

Acknowledgments

The authors would like to thank John Russo and Svetislav Maric for bringing the result stated in Theorem 5.1 to their attention. They would also like to thank the anonymous reviewers for their thorough reviews, and their detailed and insightful comments. This material is based upon works supported by the Science Foundation Ireland under Grants no. 05/YI2/I677 and no. 08/RFP/MTH1164.