Advances in Mathematics of Communications doi:10.3934/amc.2011.5.547 Volume 5, No. 3, 2011, 547–553 RESULTS OF THE ENUMERATION OF COSTAS ARRAYS OF ORDER 29 Konstantinos Drakakis School of Electrical, Electronic & Mechanical Engineering University College Dublin, Belfield, Dublin 4, Ireland Francesco Iorio Autodesk Research 210 King Street East, Toronto, Ontario M5A 1J7, Canada Scott Rickard School of Electrical, Electronic & Mechanical Engineering University College Dublin, Belfield, Dublin 4, Ireland John Walsh Department of Computer Science Trinity College Dublin, College Green, Dublin 2, Ireland (Communicated by Andrew Klapper) Abstract. The results of the enumeration of Costas arrays of order 29 are presented: except for 16 arrays out of a total of 164, all other arrays found are accounted for by the Golomb and Welch construction methods. These 16 arrays, however, cannot be considered to be new, as they were discovered in the past through a semi-empirical technique. The enumeration was performed on several computer clusters and required the equivalent of 366.55 years of single CPU time. 1. Introduction Costas arrays [3] are square arrays of dots/1s and blanks/0s, such that there exists exactly one dot per row and column (that is, they are permutation arrays), and such that a) no four dots not lying on a straight line form a parallelogram, b) no four dots lying on a straight line form two equidistant pairs, and c) no three dots lying on a straight line are equidistant. They were first proposed by J.P. Costas in 1965 in the context of SONAR detection in a GE technical report [1], and later in 1984 in a journal publication [2]. There are two known algebraic construction techniques for the production of Costas arrays: the Golomb and Welch methods. Both are based on the theory of finite fields and were introduced and proven in 1984 by S. Golomb and H. Taylor [6, 8]. These methods successfully construct n × n Costas arrays for infinitely many, but not all, orders n, and they remain the only generally applicable construction techniques for Costas arrays available today. 2000 Mathematics Subject Classification: Primary: 05A15, 05B10; Secondary: 05A05, 11B50. Key words and phrases: Costas arrays, enumeration, order 29, Golomb method, Welch method. Konstantinos Drakakis and Scott Rickard are also affiliated with the Claude Shannon Institute (www.shannoninstitute.ie), as well as with UCD CASL (casl.ucd.ie). 547 c 2011 AIMS-SDU