Geodesic
Enumeration of parallelograms in permutation matrices for improved bounds on the density of Costas arrays
Christopher N. Swanson1 ; Bill Correll, Jr.2 ; Randy W. Ho3
1Ashland University
2MDA Information Systems LLC
3Garmin International
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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A Costas array of order $n$ is an $n\times n$ permutation matrix such that all vectors between pairs of ones are distinct. Thus, a permutation matrix fails to be a Costas array if and only if it contains ones that form a (possibly degenerate) parallelogram. In this paper, we enumerate parallelograms in an $n\times n$ permutation matrix. We use our new formulas to improve Davies's $O(n^{-1})$ result for the density of Costas arrays.
DOI : 10.37236/5610
Classification : 05A05, 05B30
Mots-clés : Costas array, permutation, enumeration

Christopher N. Swanson  1   ; Bill Correll, Jr.  2   ; Randy W. Ho  3

1 Ashland University
2 MDA Information Systems LLC
3 Garmin International 
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     author = {Christopher N. Swanson and Bill Correll, Jr. and Randy W. Ho},
     title = {Enumeration of parallelograms in permutation matrices for improved bounds on the density of {Costas} arrays},
     journal = {The electronic journal of combinatorics},
     year = {2016},
     volume = {23},
     number = {1},
     doi = {10.37236/5610},
     zbl = {1333.05016},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/5610/}
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Christopher N. Swanson; Bill Correll, Jr.; Randy W. Ho. Enumeration of parallelograms in permutation matrices for improved bounds on the density of Costas arrays. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5610

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