Костас массиві: Геометриялық құрылым және қолданыстары

Set of marks on a 2d square grid such that no two pairs of marks are the same distance apart
In mathematics, a Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n×n square tiling such that each row or column contains only one point, and all of the n(n − 1)/2 displacement vectors 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, 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. The general enumeration of Costas arrays is an open problem in computer science and finding an algorithm that can solve it in polynomial time is an open research question.

The Lempel–Golomb construction takes α and β to be primitive elements of the finite field GF(q) and similarly defines if , otherwise 0. The result is a Costas array of size q − 2. If α + β = 1 then the first row and column may be deleted to form another Costas array of size q − 3: such a pair of primitive elements exists for every prime power q>2.

Generation of new Costas arrays by adding or subtracting a row/column or two with a 1 or a pair of 1's in a corner were published in a paper focused on generation methods and in Golomb and Taylor's landmark 1984 paper. More sophisticated methods of generating new Costas arrays by deleting rows and columns of existing Costas arrays that were generated by the Welch, Lempel or Golomb generators were published in 1992. There is no upper limit on the order for which these generators will produce Costas arrays.

Two methods that found Costas arrays up to order 52 using more complicated methods of adding or deleting rows and columns were published in 2004 and 2007.

Costas arrays on a hexagonal lattice are known as honeycomb arrays. It has been shown that there are only finitely many such arrays, which must have an odd number of elements, arranged in the shape of a hexagon. Currently, 12 such arrays (up to symmetry) are known, which has been conjectured to be the total number.