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Trinity College Dublin

The enumeration of Costas arrays of order 29

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Lead PI: 
Dr. Konstantinos Drakakis
Costas arrays are square arrays of 1s/dots and 0s/blanks, such that there is exactly one dot per row and column, and such that any two linear segments connecting pairs of dots have either different length or different slope. The latter property is equivalent to the following three conditions: a) no four non-collinear dots form a parallelogram; b) no four collinear dots form two equidistant pairs; and c) no three collinear dots are equidistant. Though they originated as time-frequency waveform descriptions in SONAR systems, they soon became objects of mathematical study, due to the many and interesting problems they give rise to in combinatorics, algebra, and number theory.Two algebraic construction techniques exist, based on the theory of finite fields, which successfully construct nxn Costas arrays for infinitely many, though not all, orders n. However, even in the orders where these methods are applicable, they fail to yield all existing Costas arrays there. Until today, the only available method guaranteed to yield all Costas arrays in a certain order is enumeration through exhaustive search, whereby each of the n! permutation arrays of order n is tested for the Costas property. Until today, enumeration has covered orders n<29. The aim of this project is to enumerate all Costas arrays of order 29.
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