On the number of Costas arrays as a function of array size

Abstract

2) the sum of the mean function and the amplitudewith phasefunction. However, if the physics of the problem of interest i s understood so that modeling one or both of the smooth functions with as few coefficients is possible, a generalization of Ksienski's algorithm using least squares to minimize with respect to the known functions could be much more accurate and effective.

The Status of Costas Array Construction

“…In [8], Silverman et al presented a probabilistic argument to suggest that C(n), the number of n × n Costas arrays, should go to 0 for large n. Indeed, as seen in the table of values of C(n) for 2 ≤ n ≤ 26 ( [9], p. 417), the value of C(n) increases monotonically from C(2) = 2 to C(16) = 21, 104, beyond which it decreases monotonically down to C(26) = 56. The reduced number, c(n), of n × n Costas arrays when patterns which differ only by symmetries of the square are not considered distinct, exhibits similar behavior, from C(2) = 1 up to c(16) = 2, 648 and then down to c(26) = 8.…”

Data mining and Costas arrays

“…We will proceed now to take as a working assumption that the entries of the difference triangle are reasonably uncorrelated (a more precise description of the correlation of the entries of the difference triangle is given in [1]). …”

“…We could follow the same steps we followed in the previous experiment, but there is an additional complication that must be tackled here: whereas there are exactly (n − 1)! permutations with a given f (1) This leads naturally to the need to count C(n, k); this result is standard in Combinatorics and it can be found in [10], for example, but we will derive it here, too: if we take a permutation contributing to C(n, k) and we place n + 1 in it at position i from the left, we will create a new permutation of order n + 1 contributing to C(n + 1, k + i − 1). We establish thus a recursion: if we consider a permutation contributing to C(n, k) and we remove n completely from it, we end up with a permutation contributing to…”

A Closed Form Expression for the Number of Costas Arrays of Arbitrary Order

“…Specifically, if we have 4 Costas array were first defined in [6] more than 40 years ago. Since then all Costas arrays for n < 26 have been found by exhaustive search [4], [15] and are enumerated in Table 1.2. Costas arrays are known to exist for infinitely many n due to constructions for n equal to or a little less than a prime or power of a prime. Four new Costas of orders 29, 29, 36 and 42 were found by Rickard in his application for n < 100 of 1-gap augmentation [13], a construction that attempts to create higher order Costas arrays from lower order ones.…”