Two experimental pearls in Costas arrays Konstantinos Drakakis, Rod Gow School of Mathematics and UCD CASL University College Dublin Belfield, Dublin 4 Ireland Email: {Konstantinos.Drakakis, Rod.Gow}@ucd.ie Abstract—The results of 2 experiments in Costas arrays are presented, for which theoretical explanation is still not available: the number of dots on the main diagonal of exponential Welch arrays, and the parity populations of Golomb arrays generated in fields of characteristic 2. I. I NTRODUCTION Costas arrays appeared for the first time in 1965 in the context of SONAR detection ([4], and later [5] as a journal publication), when J. P. Costas, disappointed by the poor performance of SONARs, used them to describe a novel frequency hopping pattern for SONARs with optimal auto- correlation properties. At that stage their study was entirely empirical and application-oriented. In 1984, however, after the publication by S. Golomb [9] of the 2 main construction methods for Costas arrays (the Welch and the Golomb algo- rithm) based on finite fields, still the only ones available today, they officially acquired their present name and they became an object of mathematical interest and study. Soon it became clear that the mathematical problems related to Costas arrays presented a challenge for our present method- ology in Discrete Mathematics (based on Combinatorics, Alge- bra, and Number Theory), and suggested that novel techniques are desperately needed, perhaps currently lying beyond the frontiers of our knowledge. Indeed, we have so far been unable to settle even the most fundamental question in the field: do Costas arrays exist for all orders? These insurmountable difficulties the researchers were faced with triggered inevitably an intense activity in computer exploration of Costas arrays (for example [1], [3], [14]), the rationale being that it is easier to prove something on which strong evidence has been gathered, rather than starting completely from scratch. Such evidence led to the formulation of conjectures, some of which subsequently were, at least partially, proved. These successes, however small, helped consolidate the position of the experimental method as an indispensable tool for the study of Costas arrays. Not all computer experiments have led to conjectures, however, let alone successfully proved conjectures: several experiments, perhaps the most interesting ones, yielded results that still defy any attempt for explanation. In this work we collect our findings in 2 numerical experiments we performed on Costas arrays, whose results appear very interesting, but entirely inexplicable at present, and we present them to the broader scientific community, hoping to accelerate progress towards their solution. They are: • The number of dots on the main diagonal of exponential Welch arrays; • and the parity populations of Golomb arrays generated in fields of characteristic 2. The reason for choosing these particular 2 experiments is that, having spent lots of time studying them, we can confidently say that a lot more is to be gained than mere deeper understanding of Costas arrays through their successful explanation: in our opinion, such an explanation relies on completely novel, as yet unexplored areas of finite fields, and traditional algebraic and number theoretic methods are totally incapable of making any progress. In other words, these problems, although originating in the relatively unknown field of Costas arrays, reveal new directions in Algebra and Number Theory, and are, consequently, of paramount pure mathematical interest. II. BASICS In this section we give precise definitions for all the terms used in the introduction, as well as for everything else needed in the paper. A. Definition of the Costas property Simply put, a Costas array is a square arrangement of dots and blanks, such that there is exactly one dot per row and column, and such that all vectors between dots are distinct. Definition 1. Let f :[n] → [n], where [n]= {1,...,n}, n ∈ N, be a bijection; then f has the Costas property iff the collection of vectors {(i - j, f (i) - f (j )) : 1 ≤ j<i ≤ n}, called the distance vectors, are all distinct, in which case f is called a Costas permutation. The corresponding Costas array A f is the square array n × n where the elements at (f (i),i),i ∈ [n] are equal to 1 (dots), while the remaining elements are equal to 0 (blanks): A f =[a ij ]=  1 if i = f (j ) 0 otherwise ,j ∈ [n] Remark 1. The operations of horizontal flip, vertical flip, and transposition on a Costas array result to a Costas array as well: hence, out of a Costas array 8 can be created, or 4 if the particular Costas array is symmetric.