Numerical Representation
A Costas array may be represented numerically as an n×n array of numbers, where each entry is either 1, for a point, or 0, for the absence of a point. When interpreted as binary matrices, these arrays of numbers have the property that, since each row and column has the constraint that it only has one point on it, they are therefore also permutation matrices. Thus, the Costas arrays for any given n are a subset of the permutation matrices of order n.
Arrays are usually described as a series of indices specifying the column for any row. Since it is given that any column has only one point, it is possible to represent an array one-dimensionally. For instance, the following is a valid Costas array of order N = 4:
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
There are dots at coordinates: (1,2), (2,1), (3,3), (4,4)
Since the x-coordinate increases linearly, we can write this in shorthand as the set of all y-coordinates. The position in the set would then be the x-coordinate. Observe: {2,1,3,4} would describe the aforementioned array. This makes it easy to communicate the arrays for a given order of N.
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“The terrible tabulation of the French statists brings every piece of whim and humor to be reducible also to exact numerical ratios. If one man in twenty thousand, or in thirty thousand, eats shoes, or marries his grandmother, then, in every twenty thousand, or thirty thousand, is found one man who eats shoes, or marries his grandmother.”
—Ralph Waldo Emerson (1803–1882)