1076 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013

Algebraic Symmetries of Generic-Dimensional Periodic Costas Arrays

José Ortiz-Ubarri, Member, IEEE, Oscar Moreno, Fellow, IEEE, Andrew Z. Tirkel, Senior Member, IEEE,Rafael A. Arce-Nazario, Member, IEEE, and Solomon W. Golomb, Life Fellow, IEEE

Abstract—In this paper, we present two generators for the groupof symmetries of the generic -dimensional periodic Costasarrays over elementary abelian groups: one that is definedby multiplication on dimensions and the other by shear (ad-dition) on dimensions. Through exhaustive search, we observethat these two generators characterize the group of symmetries forthe examples we were able to compute. Following the results, weconjecture that these generators characterize the group of symme-tries of the generic -dimensional periodic Costas arraysover elementary abelian groups.

Index Terms—Arrays, code division multiplexing, correlation,multidimensional signal processing, watermarking.

I. INTRODUCTION

T WO-DIMENSIONAL Costas arrays were introduced in[1] and [2] and their periodicity properties have been

studied in [3] and [4], and more recently in [5]. Good corre-lation and full periodicity properties such as those of someCostas arrays are very important for applications in digitalwatermarking [6], [7] and optical communications [8], [9].Multidimensional periodic arrays are useful for digital wa-

termarking of video, and combined video and audio media.Three-dimensional periodic arrays also have applicationsin optical communications. In a recent work, Moreno andTirkel [10] presented a multiperiodical generalization of theWelch construction over the elementary abelian group .Although their Welch construction is too sparse for digitalwatermarking applications, its multiperiodicity property allowsthe substitution of columns by periodic sequences with goodcorrelation properties.In this paper, we generate many multidimensional periodic

Costas arrays (MPCA) over elementary abelian groups.These arrays are obtained through the application of algebraicsymmetries to the generalization of the multidimensional peri-odic Welch construction. We call these symmetries: multi-plication on the dimensions and shear (addition) on the

Manuscript received June 30, 2012; accepted September 19, 2012. Date ofpublication October 24, 2012; date of current version January 16, 2013. J. Ortiz-Ubarri was supported in part by the UPR-RP FIPI funds. R. A. Arce-Nazariowas supported in part by NSF Grant CNS-0923152.J. Ortiz-Ubarri and R. A. Arce-Nazario are with the Department of Computer

Sciences, University of Puerto Rico, San Juan, PR 00931 USA (e-mail: jose.ortiz@hpcf.upr.edu; rafael.arce@upr.edu).O. Moreno is with the Gauss Research Laboratory, Inc., San Juan, PR 00931

USA (e-mail: moreno@nic.pr).A. Z. Tirkel is with the Scientific Technology Pty Ltd., Kilaben Bay, N.S.W.

2283, Australia (e-mail: atirkel@bigpond.net.au).S. W. Golomb is with the University of Southern California, Los Angeles,

CA 90089 USA (e-mail: sgolomb@usc.edu).Communicated by T. Helleseth, Associate Editor for Sequences.Digital Object Identifier 10.1109/TIT.2012.2221678

dimensions. We independently found and by analyzingthe results obtained through exhaustive exploration of some ex-amples of Welch arrays. Later, we found that these kinds of al-gebraic permutations were used on Quadratic residue arrays byBömer et al. [11], and on 2-D Costas and sonar arrays [12].We compared the results obtained from the exhaustive explo-

ration of , , and with those using the andgenerators, and observed that the group of symmetries for

these orders is characterized with the use of the algebraic gen-erators. In other words, all the MPCA found through exhaustiveexploration are generated using a combination of the symme-tries that will be presented in this study.The rest of this paper is organized as follows. Section I-A

contains the formal definitions of a generic -dimen-sional periodic Costas array over the elementary abelian group

, the definition of the Costas property, and reviews two al-gebraic symmetries of multidimensional periodic Welch Costasarrays that are used with the symmetries presented in this paperto characterize the group of symmetries of the generic MPCA.Sections II and IV present the generators used to complete thecharacterization of the group of symmetries of the generic 3-DMPCA. Sections III and V present the generalization of the gen-erators for multiple dimensions. Section VI describes the com-putational work performed. Section VII concludes this paperwith our conjectures on the existence of the generic MPCA.

A. Definitions

We begin by providing the definitions of the generic and gen-eral -dimensional periodic Costas array over the el-ementary abelian group and introduce the generalized Welchconstruction.

Definition 1: A generic -dimensional periodic Costasarray over the elementary abelian group is a permutationfunction , where means .This function has the distinct difference property: for any ,, , implies ,where the addition and subtraction operations are performed inthe corresponding abelian group.

Example 1: The following is a grid defined over :

0018-9448/$31.00 © 2012 IEEE

ORTIZ-UBARRI et al.: ALGEBRAIC SYMMETRIES OF GENERIC -DIMENSIONAL PERIODIC COSTAS ARRAYS 1077

The following is a 3-D periodic Welch Costas array over theelementary abelian group :

Remark: Through the examples of this study, the rows arenumbered upward, starting from the bottom, while the columnsare numbered from left to right.

Definition 2: A general periodic Costas array isdefined as in Definition 1, but can be found over any gridlocation.The well-known 2-DWelch construction [2] is generalized to

construct MPCA.Construction 1 (Generalized Welch: A Multiperiodic Costas

Array in Dimensions With Correlation 1): Consider thesequence such that where , andis a primitive element of . Specifically for ,

. The sequence scheme is to use the -tuplerepresentation of to determine the coordinates (location) onthe -dimensional integer grid defined above. is a periodicsequence with period . For our Costas-type construction,we take the grid point location belonging to and place a 1in a column of length located above the grid point, andzeros in all other entries in that column.The 3-D periodic Welch Costas array of Example 1 is con-

structed from modulo .

Theorem 1: The sequence posseses the Costas property:, and is, therefore,

a Costas array.Proof: Let , , and

(1)

(2)

(3)

Therefore, for .

B. Multiplication (Modulo ) and Addition (Modulo) Symmetries

The following two algebraic symmetries were introduced byMoreno et al. [13] and are an instrument for the characterizationof the group of symmetries of our examples. The multiplication(modulo ) is the first symmetry applied to the sequencesobtained from the Welch constructions; then, we apply the G1and G2 symmetries introduced in this paper, and with the addi-tion symmetry, we finish the characterization of our examples.

Theorem 2: Multiplication (modulo ) of a periodicCostas array by an integer less than and relatively prime togenerates a new periodic Costas array.

Example 2: Multiplication of by mod

Theorem 3: Addition (modulo ) of any integer lessthan to a periodic Costas array generates a new periodicCostas array.

Example 3: Addition of 4 to

The addition and multiplication (modulo ) symmetriesare used in combination with the symmetries presented in thenext sections to generate many MPCA and to characterize thegroup of symmetries of the examples that we also characterizedcomputationally.

II. ROW AND COLUMN MULTIPLICATION

For the 3-D Welch MPCA, the rows and columns multipli-cation symmetries consist in shifting the rows and columnsof its 2-D representation. In general, these symmetries consistof shifting -dimensions of the -dimensional WelchMPCA times their original position. In the following sections,we define the row multiplication (RG1), column multiplication(CG1), and the combination of the row and column multi-plication symmetries (G1), and apply them to the 3-D WelchMPCA. Finally, we define a generalization of theG1 symmetryfor dimensions.

A. Row Multiplication

Let , and , and let bethe element of matrix in position .

Definition 3 (Row Multiplication):is a permutation function such thatpermutes the rows from position to position of a 3-D peri-odic Costas over the elementary abelian group .

Example 4: The following 3-D periodic Welch Costas arrayover the elementary abelian group is one symmetry,out of four distinct rowmultiplication symmetries obtained afterapplying

1078 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013

B. Column Multiplication

Let , and , and let bethe element of matrix in position .

Definition 4 (Column Multiplication):is a permutation function such thatpermutes the columns from position to position

of a 3-D periodic Costas over the elementary abelian group.

Example 5: The following 3-D periodic Welch Costas arrayover the elementary abelian group is one symmetry,out of four distinct column multiplication symmetries obtainedafter applying the

We can combine both transformations to generate moreWelch Costas arrays.Note that .

Example 6: The following is the 3-D periodic Welch Costasarray over the elementary abelian group after applyingthe

C. Row and Column Multiplication

Definition 5 (Row and Column Multiplication):is a permutation function such that

permutes the rows from position to posi-tion and the columns from position to position of a 3-Dperiodic Costas over the elementary abelian group .In other words, .

Example 7: The following is the 3-D periodic Welch Costasarray over the elementary abelian group after applyingthe

Theorem 4: The application of the permutation functionto a 3-D periodic Costas array over the elementary

abelian group generates a new 3-D periodic Costasarray over the elementary abelian group .

Proof: Let , , , and ,,

log log (4)

log (5)

log (6)

Proof of the Costas property

log

log (7)

(8)

Therefore, for

III. GENERALIZATION: -DIMENSIONSMULTIPLICATION PERMUTATION

Let , and, such that ,

and let be the elements of -dimensionalmatrix in positions .

Definition 6 (G1: Multiplication Permutation):is a permutation function such

thatpermutes the elements of matrix from position

to position of a-dimensional periodic Costas over the elementary

abelian group .

Theorem 5: The application of the permutation functionto a -dimensional periodic Costas array over the ele-mentary abelian group generates new -dimen-sional periodic Costas arrays over the elementary abelian group

.Proof: Let , , and , such that

and let

log log (9)

log log (10)

Proof of the Costas property

log log (11)

(12)

Therefore, for

IV. ROW AND COLUMN SHEAR

The row (column) shear symmetry consists of shifting eachof the elements of the row (column) of the 2-D representation

ORTIZ-UBARRI et al.: ALGEBRAIC SYMMETRIES OF GENERIC -DIMENSIONAL PERIODIC COSTAS ARRAYS 1079

of the 3-D Welch MPCA by a different number of shifts. Ex-ample 8 shows how to obtain a new MPCA by shifting rows 0through 4 by 0 through 4 times to the right (for example, row 0is not shifted, row 1 is cyclically shifted 1 time to the right, row2 is cyclically shifted 2 times to the right, and so on). In the fol-lowing sections, we define the row shear (RG2), column shear(CG2), and the combination of the row and column shear sym-metries (G2), and apply them to the 3-D Welch MPCA. Then,we define a generalization of the symmetry fordimensions.

A. Row Shear

Let , and , and let bethe element of matrix in position .

Definition 7 (Row Shear): is a per-mutation function such that shifts theelements of the rows from position to position of a 3-Dperiodic Costas over the elementary abelian group .

Example 8: The following 3-D periodic Welch Costas arrayover the elementary abelian group is one symmetryout of five distinct column shear shift symmetries obtained afterapplying the

B. Column Shear

Let , and , and let bethe element of matrix in position .

Definition 8 (Column Shear): be apermutation function such that shiftsthe elements of the columns from position to positionof a 3-D periodic Costas over the elementary abelian group

.

Example 9: The following 3-D periodic Welch Costas arrayover the elementary abelian group is one symmetry outof five distinct column shear shift symmetries after applying the

Note that because in the permutation function, the new posi-tion in the first dimension depends on the position of the seconddimension . How-ever, the composition of these generators generates anotherMPCA.

Example 10: The following is the 3-D periodic Welch Costasarray over the elementary abelian group after applying

Let , and , , and letbe the element of matrix in position .

Definition 9 (Row and Column Shear):is a permutation function such that

shifts the elements from position to posi-tion of a 3-D periodic Costas over the ele-mentary abelian group whenever.

Theorem 6: The application of the row and column shift per-mutation function to a 3-D periodic Costas array overthe elementary abelian group generates a new 3-D pe-riodic Costas array over the elementary abelian group .

Proof: Let , , , and ,

log

log (13)

log

log (14)

Proof of the Costas property

log

log (15)

(16)

(17)

(18)

(19)

Therefore, for

1080 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013

TABLE ISOME EXPERIMENTAL RESULTS FOR 0-BASED MPCA ENUMERATIONS

V. GENERALIZATION: -DIMENSIONSSHEAR PERMUTATION

Let , , , ,and , and let be the elements of-dimensional matrix in positions .

Definition 10 (G2: Shear Permutation):is a permutation function such that

per-mutes the elements of matrix from positionto positionof a -dimensional periodic Costasover the elementary abelian group whenever

.

Theorem 7: The application of the permutation functionto a -dimensional periodic Costas array over the ele-mentary abelian group generates new -dimen-sional periodic Costas arrays over the elementary abelian group

.Proof: The proof is similar to the proof in Theorem 6.

VI. COMPUTATIONAL RESULTS

The exhaustive exploration for the enumeration of MPCAhas the same computational complexity of the well-knownproblem of enumeration of 2-D Costas arrays of size ,which is . More precisely, the exhaustive explorationfor the enumeration of periodic MPCA over the elementaryabelian group is , since we can compute all the0-based MPCAs, i.e., MPCAs with , andgenerate all the other MPCAs with the addition symmetry.At the moment of writing this paper, the last enumeration re-

ported finished for 2-D Costas arrays is for length 29 [14] andthe time per single CPUwas determined to be 70 years.With ourresources, we were able to enumerate the MPCA over elemen-tary abelian groups of order , and , which aresimilar to computing the 2-D Costas arrays of lengths 7, 23, and6, respectively. For these enumerations, the results are exactlythe same that are produced with the use of generators and. To complete the enumeration for would require ap-

proximately the equivalent to compute the enumeration of 2-DCostas arrays of length 47 (approximately 1.3 years persingle CPU). However, computing using our algebraicgenerators can be accomplished in seconds with one CPU. SeeTable I for results on the enumeration of 0-based MPCA.

A. Algorithm Using Generators G1 and G2

Our algorithm to compute the group of symmetries of thegeneric -dimensional periodic Costas arrays consists in:

1) construct all the -dimensional periodic Welch Costasarrays in . We can construct a Welch Costas array foreach primitive polynomial over . The number of prim-itive polynomials over is given by , where

is Euler’s totient function, the number of positive inte-gers less than and relatively prime to . 2) When , foreach array obtained in step 1, apply the multiplication (modulo

) symmetry to obtain MPCAs.3) For each MPCA obtained in the previous steps, construct

symmetries using generator , then 4) for eacharray obtained in the previous steps, construct symmetrieswith generator . 5) Finally, apply the addition symmetry toobtain the remaining symmetries.Note that the generators G1 and G2 are applied to thedimensions that generates 0-based MPCAs. For instance, to

compute the group of symmetries for the generic 3-D periodicCostas arrays, the generators RG1 and RG2 are applied in steps3 and 4, respectively, leaving the 0 in position (0,1). This way,we obtain all the 0-based MPCA symmetries and complete allthe other symmetries with the addition symmetry in step v.In summary, when , we construct

(20)

3-D periodic Welch Costas arrays, and for

(21)-dimensional periodic Welch Costas arrays.

B. Exhaustive Exploration Algorithm

The algorithm to compute the multidimensional examplesis similar to the algorithm to construct 2-D Costas arrays [14],[15]. It consists in a 1-D permutator that uses a backtrackingalgorithm to prune the search space. Backtracking is a generalalgorithm for solving computational problems by incremen-tally generating all possible solutions. The execution of abacktracking algorithm can be modeled as a search tree whereevery node is a partial solution. Moving forward corresponds toapproaching a valid solution, and going backward correspondsto abandoning a partial candidate that cannot possibly generatea valid solution. In this case, the possible solutions are theMPCAs, and the partial solutions are subpermutations thatmeet the multiperiodic Costas property. The main change tothe original algorithm for two dimensions is in the function tocheck whether a subpermutation meets the 3-D or 4-D Costasproperty. We have parallel implementations of these algorithmsusing MPI, CUDA (GPUs), and FPGAs.

VII. CONCLUSION

We have presented two generators and for thegroup of symmetries of MPCAs over elementary abelian group

. By applying and , we characterized the fullgroup of symmetries for the orders , and .Because of the strong properties of MPCA, perfect autocorre-lation, and multiperiodicity, we conjecture:

ORTIZ-UBARRI et al.: ALGEBRAIC SYMMETRIES OF GENERIC -DIMENSIONAL PERIODIC COSTAS ARRAYS 1081

Conjecture 1: The generators and characterize thegroup of symmetries of the MPCA over elementary abeliangroup .Conjecture 2: The number of arrays in the group of sym-

metries of 3-D periodic Costas arrays over elementary abeliangroup is given by

(22)

Conjecture 3: The number of arrays in the group of symme-tries of -dimensional periodic Costas arrays over ele-mentary abelian group for is given by

(23)

We also completed exhaustive search for the cases: ,, with empty results, and for the same reason

as above, we conjecture:Conjecture 4: There are noMPCA other than theMPCA over

the elementary abelian group .

REFERENCES

[1] J. P. Costas, “Medium constraints on sonar design and performance,”FASCON Convent. Rec. pp. 68A–68L, 1975.

[2] S. Golomb, “Algebraic constructions for Costas arrays,” J. Combinat.Theory Ser. A, vol. 37, no. 1, pp. 13–21, 1984.

[3] S. W. Golomb and H. Taylor, “Construction and properties of Costasarrays,” Proc. IEEE, vol. 72, no. 8, pp. 1143–1163, Sep. 1984.

[4] O. Moreno, S. W. Golomb, and C. J. Corrada, “Extended sonar se-quences,” IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1999–2005, Nov.1997.

[5] O. Moreno and J. Ortiz-Ubarri, “A new method to construct doubleperiodic arrays with optimal correlation,” presented at the presented atthe IEEE Inf. Theory Workshop, Taormina, Italy, Oct. 2009.

[6] A. Z. Tirkel, G. A. Rankin, R. M. van Schyndel, W. J. Ho, N. R. A.Mee, and C. F. Osborne, “Electronic water mark,” in Proc. DigitalImage Comput., Technol. Appl., Sydney, N.S.W., Australia, 1993, pp.666–673.

[7] A. Tirkel and T. Hall, “New matrices with good auto and cross-corre-lation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci., vol.E89-A, no. 9, pp. 2315–2321, 2006.

[8] S. Kim, K. Yu, and N. Park, “A new family of space/wavelength/timespread three dimensional optical code for OCDMA networks,” J.Lightw. Technol., vol. 18, no. 4, pp. 502–511, Apr. 2000.

[9] J. Ortiz-Ubarri, O.Moreno, and A. Tirkel, “Three dimensional periodicoptical orthogonal code for OCDMA systems,” in Proc. Inf. TheoryWorkshop, 2011, pp. 170–174.

[10] O.Moreno andA. Tirkel, “Multidimensional arrays for watermarking,”in Proc. Int. Symp. Inf. Theory, 2011, pp. 2691–2695.

[11] L. Bömer, M. Antweiler, and H. Schotten, “Quadratic residue arrays,”Frequenz, vol. 47, no. 7–8, pp. 190–196, 1993.

[12] O.Moreno, R. A. Games, and H. Taylor, “Sonar sequences from Costasarrays and the best known sonar sequences with up to 100 symbols,”IEEE Trans. Inf. Theory, vol. 39, no. 6, pp. 1985–1987, Nov. 1993.

[13] O. Moreno, A. Tirkel, and K. D. S. Golomb, “Multidimensional peri-odic Costas arrays over elementary abelian groups,” Preprint 2011.

[14] K. Drakakis, F. Iorio, S. Rickard, and J. Walsh, “Results of the enu-meration of Costas arrays of order 29,” Adv. Math. Commun., vol. 5,no. 3, pp. 547–553, 2011.

[15] R. Arce-Nazario and J. Ortiz-Ubarri, “Enumeration of Costas arraysin FPGAs and GPUs,” in Proc. Int. Conf. ReConFigurable Comput.FPGAs, 2011, pp. 462–467.

José Ortiz-Ubarri (M’09) received the B.Sc. degree in computer science fromthe University of Puerto Rico (UPR), Rio Piedras campus, in 2002, and his Ph.D.degree in Computing and Information Science and Engineering from the UPR,Mayagüez campus, in 2010. He was a system and network administrator, andscientific programmer (2002–2010) at the High Performance Computing facilityof the UPR. He is now an assistant professor in the Department of Computer Sci-ence, UPR, Rio Piedras Campus. His research interest include code design forsecurity and communications applications, high performance computing, andcomputer networks.

Oscar Moreno (M’81–F’99) received his Ph.D. degree from the University ofCalifornia at Berkeley in 1973. For many years he was a professor of math-ematics and computer sciences at the University of Puerto Rico, Rio PiedrasCampus. He has been the Principal Investigator of many funded projects inComputational Mathematics. He is now the founder and CEO of the Gauss Re-search Laboratory Inc., the managing organization of the Puerto Rico’s top leveldomain, which is also dedicated to research in the areas of: digital watermarking,frequency hopping communications, optical and wireless CDMA, and bioinfor-matics; where he is in the process of obtaining several patents. He has super-vised Ph.D. students in mathematics, computer sciences, and bioinformatics inthe UPR.

Andrew Z. Tirkel (M’77–SM’84) received his Ph.D. degree in physics fromMonash University, Australia, in 1975. For 5 years he was a millimeter waveengineer in the aerospace industry in the US. He taught electronics and physicsat RMIT, Australia for 5 years. Since 1986 he has been the founder and di-rector of Scientific Technology, a company performing contract R&D in sensors,instrumentation, radar, communications and electromagnetics. He has also su-pervised Ph.D. students in mathematics, physics, and electrical engineering asan honorary in 5 universities in Australia. In 1992, together with Charles Os-borne (1939–2012) he invented the digital watermark, and since then has beenstudying algebraic constructions of patterns suitable for watermarking.

Rafael A. Arce-Nazario (M’12) received the B.Sc. degree in ComputerEngineering from the University of Puerto Rico—Mayagüez (UPRM), in1992, the M.Sc. degree in electrical engineering from the University Wis-consin—Madison, in 1993, and the Ph.D. degree in Computing and InformationScience and Engineering from UPRM, in 2008. He is associate professor inthe Department of Computer Science, University of Puerto Rico—Río Piedras.His research interests include electronic design automation, design and imple-mentation of algorithms, and reconfigurable computing.

Solomon W. Golomb (M’57–F’82–LF’07) was born May 31, 1932, in Balti-more, MD. He received his B.A. in 1951 from Hopkins and his M.A. and Ph.D.degrees from Harvard in 1953 and 1957, all in mathematics. After a Fulbrightyear (1955–1956) in Norway, he joined the staff at the Jet Propulsion Labo-ratory (1956) as a Senior Research Engineer, becoming Group Leader of theInformation Processing Group (1958), and Deputy Chief of the Telecommuni-cations Research Section (1960). In 1963 he joined the faculty of the Universityof Southern California, where he holds the rank of Distinguished and Univer-sity Professor, with a joint appointment in Electrical Engineering and Math-ematics. At USC, he was President of the Faculty Senate (1976–1977), ViceProvost for Research (1986–1989), and Director of Technology—AnnenbergCenter for Communication (1995–1998). The central theme of his research hasbeen developing and applying concepts of discrete mathematics to signal designfor secure, reliable, and synchronized communications. He has published over200 technical papers, and is author or co-author of five books currently in print.He is a Fellow of the IEEE and of the AAAS. He has served on the Board ofGovernors of the Information Theory Society, and was its Shannon Lecturer in1985. He was elected to the U.S. National Academy of Engineering (1976); tothe U.S. National Academy of Sciences in 2003; and to the Russian Academyof Natural Sciences (1994) as a foreign member. His numerous awards includethree honorary doctorate degrees; the Hamming Gold Medal of the IEEE; andthe William Procter Prize of Sigma Xi.