Costas array: Difference between revisions

Revision as of 00:53, 20 December 2017

Revision as of 19:43, 21 September 2018

Line 1:

In mathematics, a '''Costas array''' can be regarded [[geometry|geometrically]] as a set of ''n'' points lying on the [[Square (geometry)|square]]s of a ''n''×''n'' [[checkerboard]], such that each row or column contains only one point, and that all of the ''n''(''n'' − 1)/2 [[displacement (vector)|displacement]] [[Euclidean vector (geometric)|vectorvectors]]s between each pair of dots are distinct. This results in an ideal '"thumbtack'" auto-[[ambiguity function]], making the arrays useful in applications such as [[sonar]] and [[radar]]. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional [[Golomb ruler]] construction, and, as well as being of mathematical interest, have similar applications in [[experimental design]] and [[phased array]] radar engineering.

Costas arrays are named after [[John P. Costas (engineer)|John P. Costas]], who first wrote about them in a 1965 technical report. Independently, [[Edgar Gilbert]] also wrote about them in the same year, publishing what is now known as the logarithmic Welch method of constructing Costas arrays.<ref>{{harvtxt|Costas|1965}}; {{harvtxt|Gilbert|1965}}; [http://nanoexplanations.wordpress.com/2011/10/09/an-independent-discovery-of-costas-arrays/ An independent discovery of Costas arrays], Aaron Sterling, October 9, 2011.</ref>

==Numerical representation==

A Costas array may be represented numerically as an ''n''&times;''n'' array of numbers, where each entry is either 1, for a point, or 0, for the absence of a point. When interpreted as [[binaryLogical matrix|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 matrix|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''&nbsp;=&nbsp;4:

Line 261:

**A008404: [[OEIS:A008404|Number of Costas arrays of order ''n'', counting rotations and flips as distinct.]]

**A001441: [[OEIS:A001441|Number of inequivalent Costas arrays of order ''n'' under dihedral group.]]

* {{springerSpringerEOM|title=Costas array|id=p/c110440}}

[[Category:Permutations]]