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Line 155: ===Lempel–Golomb=== The Lempel–Golomb construction takes α and β to be [[primitive element (finite field)\|primitive element]]s of the [[finite field]] GF(''q'') and similarly defines <math>A_{i,j} = 1</math> if <math>\alpha^i + \beta^j = 1</math>, 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''. ===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 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_4$and 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. ===Other Methods=== 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<ref>Scott Rickard, ''Searching for Costas Arrays using Periodicity Properties'', IMA International Conference on Mathematics in Signal Processing (2004}</ref> and 2007<ref>James K. Beard, Jon C. Russo and Keith G. Erickson and Michael C. Monteleone and Michael T. Wright, ''Costas array generation and search methodology'', IEEE Transactions on Aerospace and Electronic Systems, volume 43 number 2, April 2007, pp 522-538, [https://doi.org/10.1109/TAES.2007.4285351\| DOI: 10.1109/TAES.2007.4285351]</ref> == See also ==

Costas array: Difference between revisions