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 = 6×8+2×4). 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.

Open problems in Costas arrays

“…The majority of Costas arrays in the orders lying towards the interior of the lobe are sporadic Costas arrays: for example, only 16 out of the 10240 Costas arrays of order 19 are algebraically constructed (they are W 0 -arrays, to be precise). The situation is reversed towards the edges of the lobe, where there are only two sporadic ECs for n = 26 [80] (the paper actually reports three sporadic ECs, but one of them turns out to be constructible by the Rickard method), while all Costas arrays of orders n ≤ 5 are algebraically constructed.…”

“…• All Costas arrays of orders n ≤ 29 have been found through exhaustive search [6,43,44,48,80]. ; the ≥ notation is used for orders not yet enumerated, to signify that there may be more arrays in addition to the ones known so far.…”

Searching for Costas Arrays Using General Particle Swarm Optimization

“…For a given order there are two kinds of approaches to check the existence of costas arrays. They are: algebraic constructions [7] [8], and exhaustive search methods [1] [5] [9] [10] [11] [12]. Although algebraic constructions can construct infinitely many costas arrays, they do not take effect for lots of orders.…”