
TL;DR
This paper introduces three-dimensional Costas cubes, explores their existence for small orders, and presents constructions for infinite families, expanding the combinatorial understanding of Costas arrays into higher dimensions.
Contribution
It defines Costas cubes as 3D analogs of Costas arrays, analyzes their existence for small orders, and provides new infinite construction methods.
Findings
Costas cubes exist for all orders up to 29 except 18 and 19.
A significant number of Costas arrays are projections of Costas cubes.
Four infinite families of Costas cubes are constructed.
Abstract
A Costas array is a permutation array for which the vectors joining pairs of s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order Costas cube is an array of size over for which each of the three projections of the array onto two dimensions, namely and and , is an order Costas array. We determine all Costas cubes of order at most , showing that Costas cubes exist for all these orders except and and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes.
| Order | # equivalence classes | # equivalence classes | Total # |
| of Costas cubes | of Costas arrays | equivalence classes | |
| which are projections | of Costas arrays | ||
| of some Costas cube | |||
| 2 | 1 | 1 | 1 |
| 3 | 1 | 1 | 1 |
| 4 | 2 | 1 | 2 |
| 5 | 13 | 6 | 6 |
| 6 | 47 | 17 | 17 |
| 7 | 30 | 26 | 30 |
| 8 | 42 | 44 | 60 |
| 9 | 46 | 61 | 100 |
| 10 | 69 | 133 | 277 |
| 11 | 66 | 126 | 555 |
| 12 | 34 | 74 | 990 |
| 13 | 11 | 22 | 1616 |
| 14 | 6 | 6 | 2168 |
| 15 | 33 | 19 | 2467 |
| 16 | 6 | 6 | 2648 |
| 17 | 19 | 12 | 2294 |
| 18 | 0 | 0 | 1892 |
| 19 | 0 | 0 | 1283 |
| 20 | 2 | 3 | 810 |
| 21 | 50 | 20 | 446 |
| 22 | 4 | 9 | 259 |
| 23 | 11 | 7 | 114 |
| 24 | 2 | 1 | 25 |
| 25 | 20 | 7 | 12 |
| 26 | 1 | 2 | 8 |
| 27 | 77 | 27 | 29 |
| 28 | 3 | 4 | 89 |
| 29 | 33 | 18 | 23 |
| Order | # equivalence | # equivalence | # equivalence | Total # |
| classes of Costas | classes of Costas | classes of Costas | equivalence classes | |
| cubes constructed | cubes constructed | cubes constructed | of Costas cubes | |
| by TheoremĀ 7 | by TheoremĀ 9 | by TheoremĀ 11 | ||
| 2 | 1 | 1 | ||
| 3 | 1 | 1 | 1 | |
| 4 | 2 | 2 | ||
| 5 | 1 | 1 | 2 | 13 |
| 6 | 4 | 47 | ||
| 7 | 2 | 30 | ||
| 9 | 4 | 3 | 46 | |
| 11 | 4 | 3 | 66 | |
| 14 | 5 | 6 | ||
| 15 | 20 | 10 | 33 | |
| 17 | 10 | 6 | 19 | |
| 20 | 2 | 2 | ||
| 21 | 35 | 15 | 50 | |
| 23 | 10 | 11 | ||
| 24 | 2 | 2 | ||
| 25 | 20 | 20 | ||
| 27 | 56 | 21 | 77 | |
| 29 | 20 | 10 | 2 | 33 |
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Taxonomy
TopicsEconomic Theory and Policy
Costas cubes
Jonathan Jedwab
āā
Lily Yen
(16 February 2017 (revised 3 August 2017))
Abstract
A Costas array is a permutation array for which the vectors joining pairs of s are all distinct. We propose a new three-dimensional combinatorial object related to Costas arrays: an order Costas cube is an array of size over for which each of the three projections of the array onto two dimensions, namely and and , is an order Costas array. We determine all Costas cubes of order at most , showing that Costas cubes exist for all these orders except and and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes. We then present constructions for four infinite families of Costas cubes.
000 J.Ā Jedwab is with Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC V5A 1S6, Canada. L.Ā Yen is with Department of Mathematics and Statistics, Capilano University, 2055 Purcell Way, North Vancouver BC V7J 3H5, Canada and Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby BC V5A 1S6, Canada. J.Ā Jedwab is supported by NSERC. Email: [email protected], [email protected]
1 Introduction
We write for the indicator function of conditionĀ (so if is true, and 0 otherwise). Let be a permutation on . The permutation array corresponding to is the array given by , where the indices and range over . For example, representing index as increasing from left to right and index as increasing from bottom to top, the permutation array corresponding to the permutation is
1$$1$$i$$j$$2$$2$$i$$j$$3$$3$$i$$j$$4$$4$$i$$j$$5$$5$$i$$j$$6$$6$$i$$j
where 1 entries of the permutation array are represented as shaded squares.
A permutation array of order is a Costas array if the vectors formed by joining pairs of 1s in are all distinct. J.P.Ā Costas introduced these arrays in in order to improve the performance of radar and sonar systems [7]: the radar or sonar frequency is transmitted in time interval if and only if . An equivalent definition of a Costas array is a permutation array each of whose out-of-phase aperiodic autocorrelations is at mostĀ .
Each Costas array belongs to an equivalence class formed by its orbit under the action of the dihedral groupĀ (the symmetry group of a square under rotation and reflection). The equivalence class of a Costas array of order greater than 2 has size four or eight, depending on whether or not its elements have reflective symmetry about a diagonal.
In 2008, Drakakis [5] proposed a generalization of Costas arrays to dimensions other than two, based on aperiodic autocorrelations, and gave further details inĀ [6]. This viewpoint has the advantage that the one-dimensional case corresponds to a Golomb ruler [2], and was subsequently studied inĀ [1]. However, the associated generalization of a permutation to more than two dimensions is problematic when the number of dimensions is odd, and the classical constructions of Costas arrays due to Gilbert-Welch and Golomb (see TheoremsĀ 3 andĀ 4) do not seem to generalize in a natural way.
We instead propose a different generalization of Costas arrays to three dimensions, which depends directly on two-dimensional Costas arrays.
Definition 1**.**
The projections of a three-dimensional array are Projection , Projection , and Projection .
We call a multi-dimensional array whose entries all lie in an array over .
Definition 2**.**
An order Costas cube is an array over for which Projections , , are each order Costas arrays.
For example, let be the array given by
[TABLE]
Then is an order 6 Costas cube, and the Costas permutations corresponding to Projections , , are , , , respectively. The Costas cube and its three associated projections are shown in FigureĀ 1, where 1 entries of are represented by shaded cubes of size . (A three-dimensional array that is a Costas cube according to DefinitionĀ 2 is also a Costas cube according to the definition of Drakakis [5], [6]: if the multiset of vectors joining pairs of 1 entries in such an array contains a repeat, so does the multiset of vectors joining pairs of 1 entries for each of its three projections.)
Before considering Costas cubes in more detail, we explain how DefinitionĀ 2 can be formulated in terms of a three-dimensional generalization of permutation arrays. Define an order permutation cube to be an array over for which Projections , , are each orderĀ permutation arrays. Then a Costas cube is a permutation cube for which the permutation arrays given by Projections , , have the additional property that they are Costas arrays. Moreover, we can regard an order permutation array as an array over for which for each , and for each . We then see that an order permutation cube can be equivalently defined as an array over for which each two-dimensional subarray contains exactly one entry: for each , and for each , and for eachĀ . Eriksson and Linusson [10] refer to this equivalent definition of a permutation cube as a sparse 3-dimensional permutation array, and note that sparse higher-dimensional arrays were used by Pascal in 1900 [14] to define higher-dimensional determinants.
Each Costas cube belongs to an equivalence class formed by its orbit under the action of the order 48 symmetry group of a cube under rotation and reflection; the subgroup of this symmetry group under rotation but not reflection has order 24 and is isomorphic toĀ . By taking Projection (say) of each of the elements of , and discarding repeats if any, we obtain the set of distinct Costas arrays occurring as projections ofĀ . This set is the union of one or more equivalence classes of Costas arrays, and so its size is a multiple ofĀ . We may exclude reflections of when forming , because a reflection of a projection of a Costas cube can be realized as a rotation of the cube. The size of is therefore at mostĀ 24, although it can be smaller. For example, we find that as ranges over the order 6 Costas cubes (as determined by the method of SectionĀ 2), the size of the set takes each value in . In particular, the set for the order 6 Costas cube given by
[TABLE]
has size 4: its elements comprise a single equivalence class of Costas arrays whose corresponding permutations are , , , . We show in SectionĀ 3 that this example is a member of an infinite family of Costas cubes all of whose elements satisfy .
We have three principal motivations for proposing Costas cubes. The first is to provide new perspectives on the observed existence pattern for Costas arrays. The second is to ask whether the favourable projection and autocorrelation properties of Costas cubes render them suitable for use in digital communications applications such as optical orthogonal codes and digital watermarking (as has been proposed [1] for the generalization of Costas arrays due to Drakakis). The third is to present these structures as being of mathematical interest in their own right.
2 Determination of Costas cubes of order at most 29
All Costas arrays of order at most 29 have been determined by exhaustive search: those of order at most 27 were listed in the database [15], and those of order 28 and 29 are listed in [8] and [9], respectively. Now any two of the Projections , , of a permutation cube determine the cube and therefore the third Projection. We may therefore determine all Costas cubes of order in the following way. For each ordered pair of (not necessarily distinct, not necessarily inequivalent) Costas arrays (, ) of order , let be the permutation cube whose Projections and are arrays and , respectively, and retain those permutation cubes for which Projection is a Costas array. All retained permutation cubes are Costas cubes of orderĀ ; select one representative of each equivalence class of retained cubes.
TableĀ 1 displays, for each : the number of equivalence classes of Costas cubes of orderĀ ; the number of equivalence classes of Costas arrays of order which are projections of some Costas cubes of order ; and, for comparison, the total number of equivalence classes of Costas arrays of order . We see that Costas cubes exist for all orders except and , and that a significant proportion of the Costas arrays of certain orders occur as projections of Costas cubes.
3 Four infinite families of Costas cubes
In this section we give algebraic constructions for four infinite families of Costas cubes.
TheoremsĀ 3 andĀ 4 describe two classical constructions producing infinite families of Costas arrays. In these theorems (and also in TheoremsĀ 5 andĀ 6 below), the equation appearing in the argument of the indicator function is regarded over the associated field ( or ).
Theorem 3** (Gilbert-Welch construction [11], [12]).**
Let be prime, let be a primitive element of , and let . Then the array given by
[TABLE]
is an order Costas array.
Theorem 4** (Golomb construction [12]).**
Let be a prime power, and let and be (not necessarily distinct) primitive elements of . Then the array given by
[TABLE]
is an order Costas array.
Several variants of the constructions of TheoremsĀ 3 andĀ 4 have been found. Of interest in the present context are the variant family of Gilbert-Welch Costas arrays of TheoremĀ 5, as described in [4, TheoremĀ 7.30], and the variant family of Golomb Costas arrays of TheoremĀ 6.
Theorem 5** (Gilbert-Welch construction ).**
Let be prime and let be a primitive element of . Then the array given by
[TABLE]
is an order Costas array.
Theorem 6** (Golomb construction [12]).**
Let be a prime power, and let be a primitive element of for which is also primitive. Then the array given by
[TABLE]
is an order Costas array.
For each prime powerĀ , there is a primitive element for which is also primitive (as required in TheoremĀ 6) [3],Ā [13]. For each such , the Costas array satisfies and the Costas array of TheoremĀ 6 can be viewed as arising from the removal of the row and column ofĀ containing the position to form .
We now give our first algebraic construction of an infinite family of Costas cubes. Each of Projections , , in this construction is a Golomb Costas array. For a prime power, we have
[TABLE]
and
[TABLE]
Theorem 7**.**
Let be a prime power, and let , , be (not necessarily distinct) primitive elements of . Then the array over given by
[TABLE]
is an order Costas cube for which Projection is a Golomb Costas array, Projection is a Golomb Costas array, and Projection is a Golomb Costas array.
Proof.
By DefinitionĀ 2, it is sufficient to show that each of Projections , and is the specified Costas array.
Consider Projection . By (1), any two of the three conditions in the indicator function of (3) (namely and and ) imply the third, so we may rewrite (3) as
[TABLE]
For , , sum over to give
[TABLE]
Now , so by (2) there is exactly one value for which . Therefore
[TABLE]
and so Projection is a Golomb Costas array by TheoremĀ 4.
Similar arguments apply to Projections and . ā
Example 8**.**
We construct a Costas cube of order according to TheoremĀ 7, using . Represent as , and take and and . Then from (3), the triples for which are given by
[TABLE]
and the permutations corresponding to Projections , , of are given by
[TABLE]
Note that in the special case , the three projections of the Costas cubeĀ specified in TheoremĀ 7 are , and . From TheoremĀ 4, these projections all belong to a single equivalence class of Costas arrays having symmetry about a diagonal, and so the set of distinct Costas arrays occurring as projections of has sizeĀ . The order 6 Costas cube with this property that was given at the end of SectionĀ 1 is constructed by representing as and taking .
We now give our second algebraic construction of an infinite family of Costas cubes. Projections and in this construction are (equivalent to) Gilbert-Welch Costas arrays, and Projection is a Golomb Costas array.
Theorem 9**.**
Let be prime, and let , be (not necessarily distinct) primitive elements of . Then the array over given by
[TABLE]
is an order Costas cube for which Projection is a Gilbert-Welch Costas array, Projection is the reflection through a vertical axis of a Gilbert-Welch Costas array, and Projection is a Golomb Costas array.
Proof.
Consider Projection . For , sum (4) over to obtain
[TABLE]
By (2) we have , and so there is exactly one value for which . Therefore
[TABLE]
and so Projection is a Gilbert-Welch Costas array by TheoremĀ 5.
For Projection , rewrite (4) as
[TABLE]
and similarly sum over to obtain
[TABLE]
so that Projection is a Golomb Costas array.
For Projection , similarly sum (5) over to obtain
[TABLE]
The reflection through a vertical axis of a Gilbert-Welch Costas array is given by
[TABLE]
and so the reflection equals Projection . ā
Example 10**.**
We construct a Costas cube of order according to TheoremĀ 9, using , and . From (4), the triples for which are given by
[TABLE]
and the permutations corresponding to Projections , , of are given by
[TABLE]
We now give our third and fourth algebraic constructions of an infinite family of Costas cubes. Each of Projections , , in both of these constructions is (equivalent to) a Golomb Costas array.
Theorem 11**.**
Let be a prime power, and suppose there exists a primitive element of for which both and are also primitive. Then
- (i)
the array over given by
[TABLE]
is an order Costas cube for which Projection is a Golomb Costas array, Projection is a Golomb Costas array, and Projection is a Golomb Costas array. 2. (ii)
the array over given by
[TABLE]
is an order Costas cube for which Projection is a Golomb Costas array, Projection is the reflection through a vertical axis of a Golomb Costas array, and Projection is the rotation through of a Golomb Costas array.
Proof.
By DefinitionĀ 2, we must show that each of Projections , , for (i) and (ii) is the specified Costas array.
- (i)
Consider Projection . By (1), any two of the three conditions in the indicator function of (6) imply the third, so we may rewrite (6) as
[TABLE]
For , sum over to obtain
[TABLE]
Now , and is primitive by assumption, so by (2) there is exactly one value for which . Therefore
[TABLE]
and so Projection is a Golomb Costas array by TheoremĀ 6.
Similar arguments apply to Projections and . For Projection , note that we may use (1) to rewrite the condition in the indicator function of (6) as . 2. (ii)
By (1), any two of the three conditions in the indicator function of (7) imply the third. Let . By similar arguments to those used to proveĀ (i), Projections , , satisfy
[TABLE]
Therefore Projection is a Golomb Costas array by TheoremĀ 6.
The reflection through a vertical axis of a Golomb Costas array is given by
[TABLE]
and so Projection equals the reflectionĀ .
The rotation through of a Golomb Costas array is given by
[TABLE]
Therefore by (1),
[TABLE]
and so ProjectionĀ equals the rotationĀ .
ā
Example 12**.**
We construct Costas cubes and of order according to TheoremĀ 11, using . Represent as and take , for which and are also primitive. From (6), the triples for which are given by
[TABLE]
and the permutations corresponding to Projections , , of are given by
[TABLE]
From (7), the triples for which are given by
[TABLE]
and the permutations corresponding to Projections , , of are given by
[TABLE]
A primitive element for which both and are also primitive in (as required in TheoremĀ 11) does not necessarily exist for a prime power ; for example, there is no such primitive element in . If such a exists, then the order Costas cube constructed in TheoremĀ 7 with satisfies , by (1). The order Costas cube of TheoremĀ 11Ā (i) can then be viewed as arising from the removal of the three planes ofĀ containing the position to form . Furthermore, the Costas cube of TheoremĀ 11Ā (ii) can be obtained from by the rule
[TABLE]
(as illustrated in ExampleĀ 12). Application of this rule does not necessarily preserve the Costas cube property, and so and are inequivalent Costas cubes in general (even though TheoremĀ 11 shows that Projections of these two cubes are identical, Projections B are equivalent Costas arrays, and Projections C are equivalent Costas arrays).
TableĀ 2 displays, for , the number of equivalence classes of Costas cubes of orderĀ constructed by TheoremsĀ 7, 9, andĀ 11, and for comparison the total number of equivalence classes of Costas cubes of orderĀ . By direct verification, all equivalence classes of Costas cubes of order at most 29 for which all three Projections , , are Golomb Costas arrays are constructed by TheoremĀ 7; all equivalence classes of Costas cubes of order at most 29 for which two of Projections , , are equivalent to Gilbert-Welch Costas arrays and the third Projection is a Golomb Costas array are constructed by TheoremĀ 9; and all equivalence classes of Costas cubes of order greater than 2 and at most 29 for which all three Projections , , are equivalent to Golomb Costas arrays are constructed by TheoremĀ 11.
TableĀ 2 shows that the existence of Costas cubes of order 2, 3, 4, 20, 21, 24, 25, and 27 is completely explained by TheoremsĀ 7,9, andĀ 11. It would be interesting to find an explanation for the existence of the equivalence classes of Costas cubes not constructed by these three theorems, which would allow us to explain the existence of those associated Costas array projections that are currently regarded as sporadic.
Acknowledgements
We are grateful to Ladislav Stacho and Luis Goddyn, whose insightful questions and comments at the SFU Discrete Mathematics seminar in March 2017 led us to discover TheoremĀ 11. We thank the referees for their helpful comments.
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