Abstract
We describe a new way to construct finite geometric objects. For every \(k\) we obtain a symmetric configuration \(\mathcal{E }(k-1)\) with \(k\) points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.
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References
Brualdi, R.A., Pless, V.S.: Greedy codes. J. Comb. Theory, Series A 64, 10–30 (1993)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithmsm, chap. 16. In: Greedy Algorithms. MIT Press, Cambridge (2001)
Conway, J.H.: Integral lexicographic Codes. Discret. Math. 83, 219–235 (1990)
Conway, J.H., Sloane, N.J.A.: Lexicographic codes: error-correcting codes from game theory. IEEE Trans. Inf. Theory IT-32(3), 337–348 (1986)
Dembowski, P.: Finite Geometries. Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer-Verlag, Berlin, Heidelberg, New York (1968)
Edgar, T.: First-Best Projective Planes and Related Structures. Diplomarbeit, Tübingen (2009)
Euler, L.: Recherches sur une nouvelle espèce des quarrés magiques. Verh. Zeeuwsch. Genootsch. Wetensch. Vlissingen 9, 85–239 (1782)
Floyd, R.W.: Non-deterministic algorithms. J. ACM 14, 636–644 (1967)
Gropp, H.: Configurations and their realization. Discret. Math. 174, 137–151 (1997)
Hering, C.H., Krebs, A.: A partial plane of order 6 constructed from the icosahedron. Des. Codes Cryptogr. 44, 287–292 (2007)
Krebs, A., Hering, C.H., Edgar, T.: First choice constructions for non-symmetric configurations (to appear)
Lam, C.W.H.: The search for a finite projective plane of order 10. Am. Math. Mon. 98, 305–318 (1991)
ProjFinder: http://www.mathematik.uni-tuebingen.de/ab/gruppen/hering/main.html. Accessed 7 Feb 2013
MacWilliams, F.C., Sloane, N.J.A., Thompson, J.G.: On the existence of a projective plane of order 10. J. Comb. Theory 14 A, 66–78 (1973)
Ward, H.N.: A selection of divisible lexicographic codes. Int. J. Inf. Coding Theory 1(4), 410–428 (2010)
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Authors are grateful to the reviewers for many very helpful comments.
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Communicated by J. D. Key.
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Hering, C., Krebs, A. & Edgar, T. Naive configurations. Des. Codes Cryptogr. 72, 719–731 (2014). https://doi.org/10.1007/s10623-013-9797-4
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DOI: https://doi.org/10.1007/s10623-013-9797-4