Costas array: Difference between revisions

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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]] [[vector (geometric)|vector]]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>{{citationharvtxt|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>

| last = Gilbert | first = E. N. | authorlink = Edgar Gilbert

| doi = 10.1137/1007035

| issue = 2

| journal = SIAM Review

| jstor = 2027267

| pages = 189–198

| title = Latin squares which contain no repeated digrams

| volume = 7

| year = 1965}}.</ref><ref>[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==

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===Extensions by Taylor, Lempel, and Golomb===

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<ref>Solomon Golomb, ''Algebraic constructions for Costas arrays'', J. Comb. Theory Series A, volume 7 (1984), pp 1143-1163</ref> and in Golomb and Taylor's landmark 1984 paper<ref>Solomon {{sfnp|Golomb and Herbert |Taylor, ''Constructions and properties of Costas arrays'', Proceedings of the IEEE, volume 72 (|1984), pp 1143-1163</ref>}}

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<ref>Solomon W. Golomb, ''The T_4and G_4 Constructions for Costas Arrays'', IEEE Transactions on Information Theory, volume 38 (1992), pp 1404-1406.</ref>. There is no upper limit on the order for which these generators will produce Costas arrays.