The Enumeration of Costas Arrays of Size 26

Abstract

We present results of a grid computer search which enumerated the number of 26-by-26 Costas arrays. Of the 26! possible permutation matrices, only 56 of them satisfy the Costas condition that the N choose 2 line segments connecting pairs of ones are all distinct. The 56 arrays consist of 6 unique non-symmetric arrays which each generate 8 arrays using rotations and reflections and 2 symmetric arrays which each generate 4 arrays using rotations and reflections (56=6times8+2times4). This enumeration result shows the falling number of Costas arrays as a function of size continues from its peak at N=16 to N=26. As the known generation techniques produce more than 56 arrays for N=27, N=26 is the first local minima in the enumeration sequence. The search was performed on 120 machines over a three month period, and thus took nearly 30 years of CPU time. The details of how the search was performed is presented and some observations on the database of known arrays are made.