On Costas arrays with various types of symmetry. - IEEE Xplore

ON COSTAS ARRAYS WITH VARIOUS TYPES OF SYMMETRY Ken Taylor, Scott Rickard, Konstantinos Drakakis∗ Complex and Adaptive Systems Laboratory University College Dublin ABSTRACT Using the database of Costas arrays of orders 27 and below, a table is generated showing the number of arrays which exhibit some kind of symmetry. The number of diagonal, antireflective and consecutive arrays are given, correcting previously published results in a small number of cases. Finally, several of the symmetry definitions are extended to odd sizes. Index Terms— Costas Arrays, Finite Fields, Symmetric 1. INTRODUCTION Costas arrays [1, 2] are square arrangements of dots and blanks (or 1’s and 0’s, respectively) that satisfy the following two constraints: 1. Every row and column contains exactly one dot: a Costas array is, therefore, a permutation array.

kind of symmetry. This method could potentially settle the issue of the existence of Costas arrays of orders where no Costas array is yet known (the smallest such currently being 32), as a first approach to the central research question of Costas arrays today: do Costas arrays exist for all orders? 2. TYPES OF SYMMETRY This section provides the definitions of the various types of symmetry considered, along with examples. The number of Costas arrays satisfying each type of symmetry is then determined using exhaustive search. An underlying convention throughout the discussion that follows will be that a Costas array A = [aij ] of order n corresponds bijectively to the Costas permutation f such that f (j) indicates the position of the dot in column j: af (j),j = 1 and aij = 0 for i 6= f (j). 2.1. Existing Types of Symmetry

2. All vectors connecting pairs of dots must be distinct. Costas arrays first appeared in the literature as time-frequency descriptions of frequency-hopping waveforms with ideal auto-ambiguity: if two copies of a Costas array are placed on top of each other and then shifted apart by some nonzero number of rows and columns, at most one pair of overlapping dots will exist. They have applications in SONAR detection, where the ideal auto-ambiguity improves performance by reducing the occurrence of spurious position and velocity identification of targets due to noise or multi-path interference [1, 2, 3]. There exist two algebraic construction techniques for Costas arrays [4, 5, 6], based on finite fields, and known as the Golomb and Welch methods, which are applicable for infinitely many orders, but not all orders. The only known way to discover all Costas arrays for a given order is the brute-force exhaustive search, which clearly entails very high complexity and has currently been completed only up to order 27. In order to obtain a restricted yet systematic “sneak-peek” into higher orders, extra constraints can be added to the search in order to reduce the complexity (along with the scope), e.g. by focusing exclusively on permutations that satisfy a specific * The authors are also affiliated with the School of Electrical, Electronic and Mechanical Engineering, University College Dublin, Ireland

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The following three types of symmetric arrays were originally considered in [7]. Definition 1 (Diagonal) A Costas array A = [aij ] has diagonal symmetry (is diagonal) iff aij = 1 ⇔ aji = 1. Diagonal Costas arrays can be constructed using, for example, the Lempel construction [6]. A sporadic diagonal Costas array (of unexplained origin, that is) is shown in Figure 1. Definition 2 (Anti-Reflective) A Costas array A of order n = 2m has anti-reflective symmetry (is anti-reflective) iff f (i) + f (m + i) = 2m + 1 for all i. An example of an anti-reflective array is shown in Figure 2. All arrays obtained using the exponential Welch construction W1exp possess the anti-reflective property [4, 6]. Definition 3 (Consecutive) A Costas array A of order n = 2m has consecutive symmetry (is consecutive) iff |f (i) − f (n + 1 − i)| = 1 for all i. An example of a consecutive array is shown in Figure 3. This is also a sporadic array. There is no known generation method which predictably produces consecutive arrays.

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Fig. 1. A diagonal Costas array.

Fig. 3. A consecutive Costas array.

Fig. 2. An anti-reflective Costas array.

Fig. 4. A C-Anti Costas array.

2.2. New Types of Symmetry Anti-reflective and consecutive arrays are defined naturally for even orders due to the required symmetry about the vertical axis and [7] only considered then to exist for even orders. Here we consider two possible extensions to the definition of anti-reflective and consecutive arrays: C- and L-. The first option is to have the axis of symmetry run through a central column which contains a dot in the middle row. In the case of anti-reflective Costas arrays this is possible for all odd orders, but, in the case of consecutive Costas arrays, this is not the case: as there must be an even number of rows on either side of the middle row, the order n must be of the form n = 2m + 1, where m itself is an even integer. These two types of Costas arrays are then named by placing the letter C at the start to indicate that a dot was added at the center as a pivot.

Definition 4 (C-Anti) A Costas array A of order n = 2m + 1 has central anti-reflective symmetry (is C-Anti) iff f (i) + f (m + 1 + i) = 2m + 2 for all i. An example of a C-Anti array is shown in Figure 4. Definition 5 (C-Consec) A Costas array A of order n = 4m + 1 has central consecutive symmetry (is C-Consec) iff |f (i) − f (n + 1 − i)| = 1 for all i 6= 2m + 1. An example of a C-Consec array is shown in Figure 5. The second option is to move the axis of symmetry to the right by one column, or, equivalently, to move the pivoting dot to the first column instead of the center, where it still occupies the central position, and test the symmetries on all but the first column. This leads to new anti-reflective and consecutive arrays for odd orders, which are then named by placing the letter L at the start to indicate that a dot was added on the left hand side of the array.

Fig. 5. A C-Consec Costas array.

Fig. 7. A L-Consec Costas array.

types of arrays, required approximately one year of computer processing time. The database of enumerated Costas arrays currently spans all orders up to and including 27. All arrays in the database can be examined to see if they satisfy the various symmetry types, hence the database provides a fast and reliable method of checking the results of exhaustive searching. Running these checks exposed a small number of errors in the results table given in [7], and highlighted two interesting Costas arrays which are shown in Figures 10 and 11. Fig. 6. A L-Anti Costas array.

Definition 6 (L-Anti) A Costas array A of order n = 2m+1 has left anti-reflective symmetry (is L-Anti) iff f (i+1)+f (m+ 1 + i) = 2m + 2 for all i. An example of a L-Anti array is shown in Figure 6. Definition 7 (L-Consec) A Costas array A of order n = 4m + 1 has left consecutive symmetry (is L-Consec) iff |f (i + 1) − f (n + 1 − i)| = 1 for all i. An example of a L-Consec array is shown in Figure 7. 3. SEARCHING In order to find symmetric Costas arrays satisfying any of the kinds of symmetry listed above, exhaustive searching techniques were employed, combined with some extra properties Costas arrays have in order to further reduce the complexity. For example, a Costas array yields a whole family of 4 or 8 Costas arrays in total when transposed or flipped along its horizontal or vertical axis, so the searching time can be reduced by ensuring that the code will only find the version of each Costas array which comes first lexicographically within its family. The original paper on symmetric arrays [7] listed the number of arrays of each type up to order 32. Extending this table to order 36, including the addition of the new

4. THE LADENDORF METHOD All searching for symmetric Costas arrays was carried out using code which implemented the Ladendorf method 1 . Based on the fact that the lines connecting opposing corners of a parallelogram bisect each other, we can test permutations to see if they are parallelogram-free (i.e. Costas arrays) by checking that the N2 bisection points do not coincide with a dot or another bisection point. One way of doing this is to double the size of the array to include half integer points. An n × n array will require n − 1 additional rows and columns so that the intersection points of all pairs of dots may be marked in a cell. We add dots to the odd row/column positions. After each additional dot is added, we must check the midpoints of all pairs of dots to ensure that none overlap with eachother or with a dot. If an overlap occurs then the array no longer satisfies the conditions for Costas, so we remove this dot and try another location. All possible arrangements of dots are checked by recursively calling a function to add a dot to the array every time a successful addition has been made. This allows us to backtrack to a new starting point when all possibilities for a certain starting stub have been tested. Figures 8 & 9 show a graphical representation of the reasoning behind the method. 1 The

Ladendorf method was proposed by Bruce Ladendorf

Fig. 8. A non-Costas array, bisection points overlap in the center.

Fig. 10. An anti-reflective array which remains anti-reflective when transposed.

Fig. 9. A Costas array, bisection points do not overlap.

Fig. 11. A consecutive array which remains consecutive when transposed.

6. CONCLUSION 5. RESULTS

The results shown in Table 1 for orders 1–27 were obtained by testing all known Costas arrays in the Costas database [8]. The results obtained differ from the previously published ones [7] in a few instances. For trivial sizes, three omissions occurred: for example, the permutation 1342 is consecutive but was not counted in the original results. The two most interesting discrepancies, however, both occurred in order 8: it appears that there exist two Costas arrays, one anti-reflective and one consecutive, which retain their respective type of symmetry even when transposed. Since the table or results in [7] lists the number of unique arrays of each type, its values were incorrect as a result. Figures 10 and 11 show these two arrays. Compared to the previously published table, Table 1 also shows how many Costas arrays of each type are sporadic. This is important information since many of the arrays can be obtained using generation methods, for example the Lempel construction creates diagonal arrays and the Welch construction creates anti-reflective arrays. It also includes the results for the newly defined types of symmetry: the L-Anti variation yielded the largest number of arrays, while very few cases of C- and L-Consec Costas arrays were found.

The number of Costas arrays that obey some kind of symmetry was determined for orders up to and including 36, using “clever” brute-force exhaustive search. These results expand and correct some previously published results [7]. The definitions of anti-reflective and consecutive Costas arrays were extended to odd orders in two different ways and the number of arrays enumerated for all n < 36. Arrays which are both symmetric and sporadic (those not generated by the known generation techniques and their variations) only exist between sizes 10 and 23. Between 24 and 36, no symmetric arrays were found that were not already known due to the generation methods. We end with a question: Are there any symmetric Costas arrays of unknown origin which are of order greater than 23?

Acknowledgements This material is based upon works supported by the Science Foundation Ireland under Grant No. 05/YI2/I677, 06/MI/006 (Claude Shannon Institute), and 08/RFP/MTH1164. All extended results were obtained using code run on the servers of GridIreland.

N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Diag 1 1 1 1 2 5 10 9 10 14 18 (5) 17 (1) 25 (14) 23 (15) 31 (15) 20 (10) 19 (5) 10 (5) 6 (4) 4 (2) 8 (1) 5 (2) 10 (1) 0 2 2 7 0 5 4 0 0 0 1 6 1

Anti 1 2 4 3 24 (1) 44 (8) 31 (13) 77 (24) 29 3 (3) 55 0 0 84 60 0 0 108

Consec 1 1 4 9 6 4 (1) 5 (1) 8 (1) 10 (3) 3 (1) 0 0 0 0 0 0 0 0

Canti 1 0 1 0 0 0 4 (2) 0 0 0 0 0 0 0 0 0 0 0 -

Lanti 1 1 2 1 1 6 (3) 4 (3) 1 (1) 1 (1) 0 0 0 0 0 0 0 0 0 -

C-Consec 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -

L-Consec 1 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -

Symmetric Total 1 1 1 2 6 11 11 20 13 44 (1) 24 (8) 65 (10) 33 (19) 59 (29) 32 (16) 105 (35) 20 (6) 49 (8) 6 (4) 10 (6) 8 (1) 60 (2) 10 (1) 0 2 2 7 84 5 64 0 0 0 1 6 109

Table 1. Number of unique symmetric Costas arrays 1 ≤ n ≤ 36. The number in brackets shows how many of the arrays are sporadic (of unexplained origin), if any. Previously known results are shown in normal font, new results in bold, and corrected results in bold italics. All results were generated via exhaustive searching; results for n ≤ 27 were cross-checked using the database of known Costas arrays. This table corrects errors in [7] and extends the enumeration results for n > 32 and to include the odd C- and L- new symmetries. Note that the number of distinct arrays is not always the sum of the preceding columns.

7. REFERENCES [1] John Costas, “Medium Constraints on Sonar Design and Performance,” Tech. Rep. Class 1 Rep. R65EMH33, GE Co., November 1965, a synopsis of this report appeared in the Eascon. Conv. Rec., 1975, pp. 68A–68L. [2] John Costas, “A Study of Detection Waveforms Having Nearly Ideal Range-Doppler Ambiguity Properties,” vol. 72-8, pp. 996–1009, Aug. 1984. [3] Scott Rickard, “Large Sets of Frequency Hopped Waveforms with Nearly Ideal Orthogonality Properties,” M.S. thesis, MIT, 1993. [4] Konstantinos Drakakis, “A Review of Costas Arrays,” Journal of Applied Mathematics, vol. 2006, 2006. [5] Solomon Golomb, “Algebraic Constructions For Costas Arrays,” Journal Of Combinatorial Theory Series A, vol. 37, no. 1, pp. 13–21, 1984. [6] Solomon Golomb and Herbert Taylor, “Constructions and Properties of Costas Arrays,” Proceedings of the IEEE, vol. 72, no. 9, pp. 1143–1163, Sept. 1984. [7] Curtis Brown, Michal Cenki, Richard Games, Joseph Rushanan, Oscar Moreno, and Pei Pei, “New Enumeration Results for Costas Arrays,” in IEEE International Symposium on Information Theory, Jan. 1993, p. 405. [8] Konstantinos Drakakis, Scott Rickard, James Beard, Rodrigo Caballero, Francesco Iorio, Gareth O’Brien, and John Walsh, “Results of the Enumeration of Costas Arrays of Order 27,” IEEE Transactions on Information Theory, vol. 54, no. 10, pp. 4684–4687, 2008.