TOM00052 ACM (Typeset by SPi, Manila, Philippines) 1 of 31 February 23, 2011 17:22 41 Costas Arrays: Survey, Standardization, and MATLAB Toolbox KEN TAYLOR, SCOTT RICKARD, and KONSTANTINOS DRAKAKIS, University College Dublin A Costas array is an arrangement of N dots on an N-by- N grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems, communication systems, cryptography, and other areas. We present a standard- ization of notation and language which can be used to discuss Costas array generation techniques and array manipulations. Using this standardization we can concisely and clearly state various theorems about Costas arrays, including several new theorems about the symmetries of Costas arrays. We also define labels for each array (generated, emergent, and sporadic), which describe whether the array is generated using a known technique, generated using a semiempirical variation of a known technique, or of unexplained origin. A new method for obtaining emergent Costas arrays, the DRT expansion, is also given here for the first time. A MATLAB Costas array toolbox has also been developed which implements the proposed standardization. The toolbox contains a comprehensive set of functions covering Costas array generation, manipulation and classification. Categories and Subject Descriptors: G.2.1 [Discrete Mathematics]: Combinatorics—Permutations and combinations General Terms: Algorithms, Standardization Additional Key Words and Phrases: Costas arrays, finite fields, MATLAB toolbox, Welch construction, Golomb construction, Lempel construction ACM Reference Format: Taylor, K., Rickard, S., and Drakakis, K. 2011. Costas arrays: Survey, standardization, and MATLAB toolbox. ACM Trans. Math. Softw. 37, 4, Article 41 (February 2011), 31 pages. DOI = 10.1145/1916461.1916465 http://doi.acm.org/10.1145/1916461.1916465 1. INTRODUCTION Costas arrays are N-by- N matrices of zeros and ones (commonly depicted by blanks and dots, respectively), N being the order of the array, which are further limited by two constraints. The first constraint is that each row and each column must contain exactly one dot, thus the grid may be represented by a permutation of the integers 1,..., N. The second constraint is that all vectors connecting pairs of dots must be distinct. This can be tested by constructing a difference triangle for the array. An example of a Costas array is shown in Figure 1. As a result, Costas arrays have ideal auto-ambiguity prop- erties [Costas 1984]. They were originally invented in the 1960s by John P. Costas, who K. Drakakis’s research was supported by the Science Foundation Ireland under Grant Nos. 05/Y12/I677, 06/MI/006 (Claude Shannon Institute), and 08/RFP/MTH1164. Authors’ addresses: K. Taylor, UCD CASL, University College Dublin, Belfield, Dublin 4, Ireland; email: ken.taylor@ucd.ie; S. Rickard and K. Drakakis, School of Electrical, Electronic and Mechani- cal Engineering, UCD CASL, University College Dublin, Belfield, Dublin 4, Ireland; {scott.rickard, konstantinos.drakakis}@ucd.ie. Permission to make digital or hard copies part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permit- ted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from the Publications Dept., ACM, Inc., 2 Penn Plaza, Suite 701, New York, NY 10121-0701 USA, fax +1 (212) 869-0481, or permissions@acm.org. © 2011 ACM 0098-3500/2011/02-ART41 $10.00 DOI 10.1145/1916461.1916465 http://doi.acm.org/10.1145/1916461.1916465 ACM Transactions on Mathematical Software, Vol. 37, No. 4, Article 41, Publication date: February 2011.