On the generalization of the Costas property to higher dimensions

We investigate the generalization of the Costas property in 3 or more dimensions, and we seek an appropriate definition; the 2 main complications are a) that the number of ``dots'' this multidimensional structure should have is not obvious, and b) that the notion of the multidimensional permutation needs some clarification. After proposing various alternatives for the generalization of the definition of the Costas property, based on the definitions of the Costas property in 1 or 2 dimensions, we also offer some construction methods, the main one of which is based on the idea of reshaping Costas arrays into higher-dimensional entities.


Introduction
Costas arrays originated in engineering as a frequency hopping pattern that optimizes the performance of radars and sonars [3,4]; being a singular combinatorial object, however, they have been lately the focus of intensive study by mathematicians as well, and have thus started leading a second independent "life" in the mathematical literature [7,8,9,10,14].In this work, we increase the level of mathematical abstraction by investigating analogs of Costas arrays in 3 or more dimensions: several challenges lie ahead, as we first need to give a satisfactory definition of the Costas property in higher dimensions, and also provide some algorithms to construct such Costas cubes or "hypercubes" in general.
We will begin by stating the definition of Costas arrays [5] in such a way as to exhibit their direct relation to Golomb rulers [16], which will consequently be construed as the 1-dimensional analog of Costas arrays.Subsequently, we will discuss several slightly different candidate definitions in higher dimensions, and we will attempt to generalize the generation methods available to us.We shall see, in particular, that it is possible to generate sparse Costas hypercubes systematically, using various construction methods, but if we want to know what the densest Costas hypercube can be, things become less clear.

Vector permutations
It is quite simple to generalize permutations in higher dimensions, if the number of dimensions is even: Definition 7 (Permutation Costas hypercube).Let m = 2s, s ∈ N, and let g : [n] s → [n] s be a bijection, that is a permutation on vectors in general.Let f : Z m → {0, 1} be a sequence such that f (i) = 1 iff (is+1, . . ., i2s) = g(i1, . . ., is), (i1, . . ., is) ∈ [n] s , and such that it has the Costas property, as defined in Definition 5; then, f will be called a permutation Costas hypercube in m dimensions with side length n.

Remark 2.
• Permutation Costas hyper-rectangles are defined by the obvious extension of the above definition.
• The fact that we chose the first s dimensions to form the domain of g and the last s its range does not affect generality: if f is a Costas hypercube, then any f ′ resulting by a random permutation of the order of the dimensions is also a Costas hypercube; this constitutes a generalization of the invariance of the Costas property in 2 dimensions under transposition.
• No 2 vectors in the collection {(i − j, g(i) − g(j)) : i, j ∈ [n] s , i = j} can be equal; however, they may have coordinates equal to 0. Remark 3. When m = 2s + 1, s ∈ N, it is clearly impossible to define a permutation as we did above; the best we can aim for is to find an injective function g :

Strict Costas hypercubes
Vectors between dots in a permutation Costas hypercube are allowed to have 0 coordinates, in contrast to ordinary 2-dimensional Costas arrays where this does not happen (see Remark 1).We could potentially restrict the definition of a Costas hypercube to exclude such a possibility, although such a restriction is extremely severe: invoking the Pigeonhole Principle, if a hypercube of side length n has n + 1 dots, then, fixing any coordinate, at least 2 of those have position vectors with the same value for the chosen coordinate, hence the corresponding distance vector has a 0 there.Therefore, our requirement limits the number of coordinates/dimensions to at most n, implying that one of the coordinates (of an element of the hypercube equal to 1) determines unambiguously the rest of them.
Definition 8 (Strict Costas hypercubes).Let m, n ∈ N and let f : and where all vectors in the family {ij − i k |1 ≤ j < k ≤ n} are distinct and have no coordinates equal to 0; then, f will be called a strict Costas hypercube.
We proceed to give 2 explicit construction methods for strict Costas hypercubes.
Proof.The result is practically obvious: a typical distance vector is (j −k, g1(j)−g1(k), . . ., gm−1(j)−gm−1(k)), where 1 ≤ j < k ≤ n, and as all gs are permutations, no coordinate is equal to 0. Further, the first 2 coordinates of the vectors above are all distinct, as g1 is a Costas permutation.Proof.Every column of A contains each integer in [n] exactly once, hence no distance vector has a coordinate equal to 0. Further, given the difference between 2 rows, the 2 rows can be uniquely determined.Indeed, if ri and rj denote the rows i and j of A, respectively, with i < j, then ri − rj is a vector with the first i coordinates equal to i − j, the following j − i coordinates equal to n + i − j, and the remaining coordinates equal to i − j; hence, i and j is uniquely determined by inspection.
Example 2. Table 3.3 shows the corresponding array A of Costas-Toeplitz hypercube with n = 4 and m = 5.
The former method has the drawback that it requires that Costas permutations of order n be known, while the latter has the drawback that m ≥ n, so in general the number of dimensions m needs to be very high; on the other hand, it requires no Costas permutations of order n to be known, so it can be used for values of n such as 32 and 33, where no Costas arrays are known yet.In any case, strict Costas hypercubes have lots of "empty space", namely an extremely low density of dots (they only have n dots), and therefore they tend not to be very interesting.

The main construction method
It is natural to ask whether Costas arrays (in 2 dimensions) can somehow be manipulated (essentially reshaped) to produce Costas hypercubes, and hopefully permutation Costas hypercubes.We prove below that this is possible, and provide several variants of this construction.

Reshaping
We formulate and prove below a general result about constructing a Costas hyper-rectangle out of a Costas square.In the special case where the Costas square is a Costas array, and the hyper-rectangle a hypercube, it turns out the hypercube is a permutation Costas hypercube.
1. Choose 2 values for i, say i1 and i2; the corresponding distance vector is: We need to show that all of these vectors are distinct.In other words, we need to show: In the last step we used the fact that g is Costas.Further, observe that the left half row vectors so constructed are the expansions of i ∈ [n] − 1, while the right half vectors are the expansions of g(i), i ∈ [n] − 1; the fact that every i ∈ [n] − 1 gets expanded exactly once and that g is a permutation guarantees that the hyper-rectangle produced is a permutation one.This completes the proof.
The hypercube of even dimension is just a special case: Corollary 1 (Costas hypercube of even dimension).Let m, n ∈ N * , and let g be a Costas permutation of order n m , but following the convention that g : Then, the hypercube of side length n whose dots (n m in total) lie at the points ( Remark 4. Notice that these hypercubes have n m dots, which is the square root of the n 2m total positions available in the hypercube, just like in Costas arrays, where there are n dots among the n 2 available positions.
Here is an attempt to construct approximate Costas hypercubes of odd dimension in the special case where the side length n is a perfect square, using Theorem 3, by first constructing a hyper-rectangle as an intermediate step: √ n ∈ N, and let g be a Costas permutation of order n m √ n, but following the convention that g : similarly, g(i) gets mapped bijectively to V (g(i)) = (v0(g(i)), v1(g(i)), . . ., vm(g(i))).
This process forms a Costas hyper-rectangle in 2m + 2 dimensions, whose side length in 2m dimensions is n and in the remaining 2 dimensions √ n.Now, replace the coordinate pair (v0(i), v0(g(i))) in the coordinate vector of each dot by the single coordinate Then, the hypercube of side length n whose dots (n m √ n in total) lie at the points However, simulations show that the damage this does to the Costas property is usually small: tests with Costas arrays of side length n 3 ≤ 200 showed that systematically over 95% of the difference vectors among the dots are distinct.
The construction methods above can be significantly extended if we use Costas squares instead of Costas arrays as the starting point.The key observation (through the proof of Theorem 3) is that the permutation property of the original Costas array is not responsible for the Costas property of the hyper-rectangle produced, but rather for its permutation property alone (and in the case of Heuristic 1 it does not even achieve that).Therefore, if we are not interested in obtaining a permutation Costas hyper-rectangle as the final product, or if a suitable sized Costas array is not available, we may as well start with a Costas square.
A special type of Costas squares that proves very helpful in practice is smaller Costas arrays.Consider a Costas array of order n ′ ∈ N * and let n > n ′ ; then, this Costas array can be turned, by the addition of n − n ′ blank rows and columns at the sides of the array, into a Costas square of size n.Note that such Costas squares are generated by incomplete permutations.We generalize this notion in the following definition: Definition 9 (Incomplete Costas array).A Costas square with the property that there is at most one dot per row and column will be called an incomplete Costas array.A (Costas) hyper-rectangle/hypercube of even dimension with the property that there are no 2 dots whose position vectors have the same left half or right half part will be called an incomplete (Costas) hyper-rectangle/hypercube.

Corollary 2 (Constructions out of Costas squares).
• A construction according to Theorem 3 starting with a (incomplete) Costas square results to a (incomplete) Costas hyper-rectangle.
• A construction according to Corollary 1 starting with a (incomplete) Costas square results to a (incomplete) Costas hypercube.
• A construction according to Heuristic 1 starting with a (incomplete) Costas square results to a hypercube that very nearly has the Costas property and usually can be turned into a Costas hypercube through the removal of a few dots.

Construction examples
We  = 32 dots, one less than that to be exact.

Applicability of reshaping
The construction method we described reshapes a Costas array into a high-dimensional Costas hypercube.As the order n m is bound to be quite big, the only Costas arrays that will normally be available are Golomb and Welch constructions [5].For practical purposes, we can seek Costas arrays of large orders in databases, such as the database created by Dr. J. K. Beard containing all known Costas arrays up to the order 200 [2].But more generally, in order to figure out whether a 2m-dimensional cube of side length n exist, we will need to check whether one of the following equations holds: • n m + 1 = p (W1 construction possible) where in all cases p is a prime.Other variants of the Golomb and the Welch construction do not occur systematically, so we do not investigate them.We now look briefly into each one of the above equations.As a general comment, the solution of these equations falls under the scope of Diophantine Analysis, and it appears that many conjectures can be formulated, but few facts have actually been proved.Assume that m = m1m2, where m1 is odd; then n m 2 + 1|n m + 1, hence it cannot be a prime.It follows that m = 2 k , for some k, whence p = n 2 k + 1.Further, n must necessarily be even, whence n = 2l and p = 2 2 k l 2 k + 1.For l = 1, we obtain p = 2 2 k + 1, the celebrated Fermat primes, for which a lot is known; in particular, it is conjectured that the only such primes correspond to 1 ≤ k ≤ 4, and lead to hypercubes of side length 2 in 2 2k dimensions.
Solutions for higher k can also be found.For example, for 1 < n ≤ 20 and k = 2, n 4 + 1 is prime for n = 2, 4, 6, 16, 20.Those solutions can actually build several hypercubes, namely either of side n 2 in 4 dimensions, or of side n in 8 dimensions, in all cases with n 4 dots.

Older generalization attempts
Extensions of the Golomb and the Welch constructions to 3 dimensions were investigated in the past [6], but the objective then was slightly different: the cubes were so constructed that all the 2-dimensional "slices" along some of their directions be Costas arrays, or at least almost Costas arrays in a certain sense.In other words, although the construction was 3-dimensional, the Costas property was still investigated in 2 dimensions.It turns out that many "reasonable" generalizations yield cubes with with 2-dimensional (almost) Costas slices.For example, the dots in the cube are placed at the points whose coordinates are the solutions (i, j, k), i, j, k ∈ [q − 1], of one of the following equations: 1. a i+x + b j+y + c k+z = 0, where a, b, c are primitive roots of the field F(q), for some q > 2 power of a prime, and x, y, z fixed in the range 0, . . ., q − 2; 2. a i+x + b j+y ≡ k mod p, where p prime, a, b primitive roots of the field F(p), and x, y fixed in the range 0, . . ., q − 2; 3. a i+j ≡ k mod p, where p prime, and a primitive root of the field F(p); 4. a i+d ≡ jk mod p, where p prime, a primitive root of the field F(p), and d fixed in the range 0, . . ., q − 2.
Note that the first one can be expanded in an obvious way into arbitrarily many dimensions.Unfortunately, computer simulations show that none of these constructions yields a Costas cube, so they are not suitable for our purposes.

An extension of the Welch construction 5.1 The original method and its non-extendability for Costas arrays
The Welch construction method for Costas arrays [5,7] stipulates that, if p is a prime, g a primitive root [1] of F(p) and c ∈ [p − 1] − 1 a fixed parameter, then the function f (i) = g i−1+c mod p, i ∈ [p − 1] is actually a Costas permutation on [p − 1].Contrary to the Golomb construction [5,7], though, which works in all finite fields, namely with p m elements where p is a prime and m ∈ N * , the Welch method is not applicable when m > 1.
The reason for that is simple: when m > 1, the elements of the field are no longer represented by integers, but rather by polynomials of degree m − 1 in a (for all practical purposes) "dummy" variable, say x, and with coefficients in [p] − 1, while addition and multiplication are no longer defined modulo an integer, but rather modulo a monic irreducible polynomial P (x) of degree m [1,5].It follows that the function f of the Welch construction is now f : [q − 1] → F * (q) where f (i) = g i−1+c mod P (x), i ∈ [q − 1], with c ∈ [q − 1] − 1 and g a primitive root of F(q) (it can be shown that the multiplicative subgroup F * (q) of the field F(q) is still cyclic [1,5]); therefore, f (i) is a polynomial, while i is an integer, and the whole construction is (at first sight, at least) meaningless, as we need f to produce integer values!Perhaps a more correct way to think of the variable x is that it represents an algebraic element of F(p) of order m; that is, it is a root of the polynomial P (x) with coefficients in F(p) and of degree m, while it is the root of no other (non-zero) polynomial with coefficients in F(p) and of degree less than m [1].
Although this construction fails to yield a Costas array, it still produces Costas hyper-rectangles, as we are about to see.

Construction of hyper-rectangles
The field F(p m ) can be construed to be a vector space over the field F(p) in 2 ways, depending on whether we consider its elements to be polynomials or m-tuples: Definition 10.The field F(p m ), where p prime and m ∈ N * , when viewed as a vector space over the field F(p), will be denoted by F(p) m .Let VP (p, m) denote the vector space of polynomials of degree m−1 over the field F(p).Then, VP (p, m) and F(p) m are isomorphic vector spaces, as they have the same finite dimension m and they are over the same field [1], under the isomorphism denoted by F, Theorem 4 (Welch hyper-rectangles and hypercubes).Let p be a prime, m ∈ N * , g a primitive root of F(q) where q = p m , and c ∈ [q − 1] − 1.We choose VP (p, m) as our representation of F(q).Then: • The hyper-rectangle in m + 1 dimensions with side length q − 1 in the first dimension and p in the others, whose dots lie at the positions with coordinates {(i, f (i))|i ∈ [q − 1]}, has the Costas property.
• The hypercube in 2m dimensions with side length p, whose dots lie at the positions with coordinates Proof.The proof really consists of putting together bits we have already proved.
• f is a permutation over F * (q) because g is taken to be a primitive root of F(q).
• That the family of vectors {(i, f (i))|i ∈ [q − 1]} has the Costas property follows from a verbatim repetition of the classical argument for m = 1 [5,7].Let us see it here in detail: consider the 4 integers i1, i2, i3 = i1 + k, and i4 ).This can be written as: , as 0 < i1 < i2 < q, which makes the second condition always false Hence, i1 = i3, i2 = i4 and the proof is complete.
• We need to show that, given that the family {(i, f (i))|i ∈ [q − 1]} has the Costas property, the family ]} has it too; but this is a verbatim repetition of the argument presented in the proof of Theorem 3.
• The hypercubes constructed above have q − 1 dots out of q 2 possible dot positions, and thus follow approximately the square root rule we saw earlier for the density.
• They are "almost" permutation hypercubes, except that the zero vectors are missing.If we set f (0) = 0 and add a dot at the position (V (0), f (0)) = (0, 0) (obviously V (0) = 0 as well), then we get a Costas hypercube but we have no guarantee anymore that it has the Costas property; simulations show that sometimes it does.This is the equivalent of the W0 construction of Costas arrays by the addition of a "corner dot" to a W1-constructed array [5,7,8].
• Welch arrays retain the Costas property when their columns get shifted circularly [5,7]; this shift is expressed by the fixed parameter c in the definition of the permutation f shown earlier.As the extension of Welch method formulated in Theorem 4 preserves this parameter c in the definition of f , we see that the hypercubes and the hyper-rectangles it produces have a similar periodicity: the fact that the family of dots at the locations {(i, f (i))|i ∈ [q − 1]} defines a Costas hyper-rectangle implies that the family of dots at the locations , also defines a Costas hyper-rectangle; similarly, the fact that the family of dots at the locations } defines a Costas hypercube implies that the family of dots at the locations , also defines a Costas hypercube.
• Just like we used V to obtain a Costas hypercube out of the originally constructed hyper-rectangle, we can use V −1 to obtain a permutation array of order q − 1, with dots at the positions {(i, The question arises naturally whether this is actually a Costas array; alas, simulations show that it isn't.
The reason is that V −1 does not generally preserve the Costas property, as we already noted in Remark 5.
• The most important aspect of this method is that it builds hypercubes not derived by Costas squares or arrays, at least in an obvious way.At the risk of sounding overly optimistic, if a method that converts Costas hypercubes into Costas arrays were available, it could potentially lead to novel Costas arrays when applied on these hypercubes.

A further generalization
The isomorphism F between the 2 representations of F(p m ) as a vector space we suggested in Section 5.2 is probably the most obvious one, but by no means the only one possible.In terms of basis correspondence, note that V (p, m) is equipped with the natural basis of polynomials Pi(x) Theorem 5 (Welch hyper-rectangles and hypercubes under arbitrary bases).Let p be a prime, m ∈ N * , g a primitive root of F(q) where q = p m , c ∈ [q − 1] − 1, and B an invertible matrix with elements in F(p), so that its rows define a basis for the vector space F(p) m over F(p).We choose VP (p, m) as our representation of F(q).Then: • The function fB : [q −1] → F * (q), where fB(i) = F(g i−1+c mod P (x))B −1 , i ∈ [q −1], P (x) an irreducible polynomial over F(p) of degree m, is a permutation over F * (q).
• The hyper-rectangle in m + 1 dimensions with side length q − 1 in the first dimension and p in the others, whose dots lie at the positions with coordinates {(i, fB(i))|i ∈ [q − 1]}, has the Costas property.
• The hypercube in 2m dimensions with side length p, whose dots lie at the positions with coordinates {(V (i), fB(i))|i ∈ [q − 1]}, has the Costas property; V is the familiar mapping from Corollary 1, with n = p in the present case. Proof.
• fB is a permutation over F * (q) because g is taken to be a primitive root of F(q) and FB is a bijection.
• It is clear that the ability to change the basis while preserving the Costas property increases tremendously the number of possible Welch-constructed Costas hypercubes.
• All these hypercubes are "almost" permutation hypercubes: the addition of one dot at the zero position vector turns them into permutation hypercubes.

Normal bases of fields and "rotational" constructions
It is well known that in every finite field F(q), q = p m , p prime, m ∈ N * there exists an element b such that the elements b p i , i = 0, . . ., m − 1 are linearly independent over F(p), thus forming a basis of F(p) m [12].In the context of our generalized Welch construction, these bases (known as normal bases) lead to nice "rotational" hypercubes.The reason is the equivalence:

Examples
Example 6.Let p = 3 and m = 3, so that q = 27, choose P (x) = x 3 + 2x + 1 which is irreducible over F(3), and choose c = 1, g = x.The Costas hyper-rectangle and the Costas hypercube constructed by Theorem 4 are shown in Table 6, along with the corresponding Welch permutation discussed in Remark 6, which fails to have the Costas property.In this particular example, adding a corner dot at (0, 0) to the hypercube preserves the Costas property, thus yielding a permutation Costas hypercube.
Example 7. Let p = 5 and m = 2, so that q = 25, P (x) = x 2 + x + 2 which is irreducible over F (5), and c = 1, g = 2x.Further, choose the array B of Theorem 5 to be The Costas hyper-rectangle and the Costas hypercube constructed by Theorem 5 are shown in Table 7, along with the corresponding Welch permutation discussed in Remark 6, which fails to have the Costas property.In this particular example, adding a corner dot at (0, 0) to the hypercube preserves the Costas property, thus yielding a permutation Costas hypercube., c = 1, P (x) = x 3 + 2x + 1 (left), the same hyper-rectangle after changing the basis using B chosen as in Example 8 (center), and the corresponding Welch hypercube (right), where we rearranged the order of the rows to demonstrate clearly the rotational structure Example 8. Let p = 3 and m = 3, so that q = 27, choose P (x) = x 3 + 2x + 1 which is irreducible over F(3), and choose c = 1, g = 2x 2 .It follows that g 3 = 2x 2 + 2x + 2, g 9 = 2x 2 + x + 2, and one can see immediately that the 3 vectors (g, g 3 , g 9 ) are linearly independent.The matrix B corresponding to this choice of a (normal) basis is: The Costas hyper-rectangle and the Costas hypercube constructed by Theorem 5 are shown in Table 8.In this particular example, adding a corner dot at (0, 0) to the hyper-rectangle (either before or after the change of basis) does not preserve the Costas property, but adding it to the hypercube does, thus producing a permutation Costas hypercube.Hence, Costas arrays became special cases of sequences with the Costas property in 2 dimensions, which we named Costas rectangles and squares; while the former generalized naturally to Costas hyper-rectangles and hypercubes, we generalized the latter in 2 distinct ways: • Strict Costas hypercubes: here, no 2 dots are allowed to have corresponding coordinates with the same value.We proposed 2 construction methods for such hypercubes, but we saw that the requirements of the definition severely limit the number of possible dots, and thus this case tends not to be very interesting.
• Permutation Costas hypercubes: here, the hypercube was viewed as the representation of a permutation between vectors instead of integers.
For the case of permutation Costas hyper-rectangles and hypercubes of even dimension, we proposed a construction method that reshapes an existing Costas array of suitable order into the desired hyper-rectangle or hypercube.In the case of odd dimension, we proposed a related heuristic that does not always work, but even when it doesn't it usually produces a hyper-rectangle that very nearly has the Costas property.We also generalized the constructions by starting with Costas squares instead of arrays, and gave specific examples.
We subsequently investigated the application of the Welch construction method on finite fields with a nonprime number of elements, and found out that, although the method fails to produce Costas arrays, it produces Costas hyper-rectangles and hypercubes in a natural way, and actually in very large numbers, due to the possibility of changing the basis of the representation while retaining the Costas property.Once more, we supplied specific examples of the construction.
Finally, we investigated experimentally, through Monte Carlo simulations, the difficult question of the restrictions that the Costas property alone imposes on the number of dots in a Costas hypercube, and we generated directly Costas hypercubes and Costas squares of various side lengths in several different dimensions.We observed that, although in small side lengths the simulations produced hypercubes more densely packed with dots than those produced by the construction methods, as the side length increased this ceased to be the case.
There are still many possible directions for future research in Costas hypercubes.For example, 1.What is the maximum number of dots that can be packed into a Costas hypercube or hyper-rectangle, given the number of dimensions and the side lengths?
2. Heuristic 1 seems to be producing Costas hypercubes pretty often, although in many occasions it fails to do so.Can we provide a rigorous and simple sufficient condition for the resulting hypercube to have the Costas property?
3. Is there a different construction method for Costas hypercubes?In particular, can the Welch and Golomb methods be generalized in a direct way (that is, without the intermediate step of the use of the mapping V defines in Theorem 1) in 3 or more dimensions?More generally, can a construction method be found directly based on finite fields?
4. Are there any engineering applications of Costas hypercubes, perhaps of a similar nature to the applications of Costas arrays?

Table 2 :
A Toeplitz construction of a Costas hypercube with m = 5, n = 4 falls within this class of vectors.

3 .
Putting things together: ( is usually a good approximation of a Costas hypercube, and the removal of a few dots turns it into a Costas hypercube.Remark 5.The reason why this heuristic often fails to produce a Costas hypercube is that different pairs of coordinates (in difference vectors) can collapse to the same value: for example, assume n = 25 ⇔ √ n = 5 and consider the pairs (−3, 0) and (2, −1); they get mapped to −3 + 5 • 0 = −3 and 2 − 5 = −3.In the context of the proof of Theorem 3, this is equivalent to saying give 3 examples: the first is the construction of a Costas hypercube with n = 5, m = 4 out of a Costas array of order 25 (using Corollary 1); the second is the construction of a Costas hypercube with n = 9, m = 3 out of a Costas array of order 27 (using Heuristic 1); and the third is the construction of an incomplete Costas hypercube with m = 5, n = 4 (using Heuristic 1), starting with a Costas array of order 31 and extending it into an incomplete Costas array of order 32.Example 3. Consider the permutation of order 25 appearing on

Table 3 :
The conversion of a Costas array of order 25 into a Costas hypercube with m = 4, n = 5: the permutation (left), and the final Costas hypercube (right) The conversion of a Costas array of order 31 into a Costas hypercube with m = 5, n = 4, treating the Costas array as an incomplete Costas array of order 32: the (incomplete) permutation (left), the intermediate hyper-rectangle (center), and the final Costas hypercube (right) 4.3.1 n m + 1 = p has the Costas property; V is the familiar mapping from Corollary 1, with n = p in the present case.

Table 1 :
A hypercube with m = 3, n = 4 constructed by permuting all pairs of integers Example 1.As a specific example, let us use m = 4, n = 3.Then, the hypercube with f (i) = 1 iff i is one of the row vectors of Table3.2 is a permutation Costas hypercube; this can be checked by a) verifying that the 2 first columns contain all vectors with integer coordinates between 1 and 3, as do the 2 last columns, and b) by != 36 distance vectors and observing they are indeed distinct.Observe also that this hypercube has in total 9 = 3 2 nonzero elements out of 81 = 3 4 ; in general, permutation Costas hypercubes have n s nonzero elements out of n 2s .

Table 3
[4]tas hyperrectangle in 6 dimensions of side lengths 4 and 2, whose dots lie at the points shown in Table5(center).Subsequently, we combine the middle columns that have values in the range {0, 1} into a single column with values in the range[4]− 1, as described in Heuristic 1.The result appears in Table5(right).A check of the Costas property shows that this time we get lucky and that the hypercube of m = 5, n = 4 we have created is Costas.It is obviously not a permutation Costas hypercube, as this term is meaningless in odd dimensions.It also has just less than √ 4 5 (left).Applying Corollary 1, we get a Costas hypercube with m = 4, n = 5, whose dot positions appear onTable 3 (right).Observe that this is a permutation Costas hypercube, as Corollary 1 states: every vector (i, j), i, j ∈ [5] − 1 appears in the left 2 columns in exactly 1 row and in the right 2 columns also in exactly one row.Example 4. Consider the Costas permutation of order 27 appearing on Table 4 (left).Applying Heuristic 1, we first get a Costas hyper-rectangle in 4 dimensions of side lengths 9 and 3, whose dots lie at the points shown in Table 4 (center).Subsequently, we combine the middle columns that have values in the range {0, 1, 2} into a single column with values in the range [9] − 1, as described in Heuristic 1: for example, the middle 2 coordinates (1, 2) in row 12 become 1 + 2 • 3 = 7.The result appears in Table 4 (right).A check of the Costas property shows that this time we get lucky and that the hypercube of m = 3, n = 9 we have created is Costas.It is obviously not a permutation Costas hypercube, as this term is meaningless in odd dimensions.Example 5. Consider the Costas permutation of order 31 appearing on Table 5 (left), and consider it as an incomplete Costas permutation of order 32 = 2 5 = 4 2 √ 4. Applying Heuristic 1, we first get a

Table 4 :
The conversion of a Costas array of order 27 into a Costas hypercube with m = 3, n = 9: the permutation (left), the intermediate hyper-rectangle (center), and the final Costas hypercube (right)

Table 7 :
The process of constructing a Welch hypercube in F(25): the Welch hyper-rectangle corresponding to g = 2x, c = 0, P (x) = x 2 + x + 2, B chosen as in Example 7 (left), the corresponding Welch hypercube (center), and the corresponding Welch permutation (right)

Table 8 :
The process of constructing a Welch hypercube in F(27): the Welch hyper-rectangle corresponding to g = 2x 2

Table 9 :
The number of dots in some Costas hypercubes constructed by means of Algorithm 1 (left), as well as in some Costas squares (right).The ticked simulations (left) are produced by running Algorithm 1 not on the full list of position vectors, but only on those that satisfy the Golomb generalization in Section 4.4.