Abstract
We derive a new existence condition for Hadamard matrices with circulant core, in terms of resultants, Hall polynomials and cyclotomic polynomials. The derivation of this condition is based on a formula for the determinant of a circulant matrix and properties of resultants.
Keywords
This research is partially supported by NSF, AirForce and NSERC grants.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Agaian, S.S.: Hadamard Matrices and their Applications. Lecture Notes in Mathematics, vol. 1168. Springer, Berlin (1985)
Apostol, T.M.: Resultants of cyclotomic polynomials. Proc. Amer. Math. Soc. 24, 457–462 (1970)
Djokovic, D.Z.: Hadamard matrices of order 764 exist. Combinatorica 28(4), 487–489 (2008)
Fee, G., Granville, A.: The prime factors of Wendt’s binomial circulant determinant. Math. Comp. 57(196), 839–848 (1991)
Hadamard, J.: Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17, 30–31 (1893)
Hall, M.: Combinatorial Theory. Wiley Classics Library. Wiley, New York (1998)
Horadam, K.J.: Hadamard Matrices and their Applications. Princeton University Press, Princeton (2007)
Kotsireas, I.S., Koukouvinos, C., Seberry, J.: Hadamard ideals and Hadamard matrices with circulant core. J. Combin. Math. Combin. Comput. 57, 47–63 (2006)
Lehmer, D.H.: Some properties of circulants. J. Number Theor. 5, 43–54 (1973)
Lidl, R., Niederreiter, H.: Finite fields. Encyclopedia of Mathematics and its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1997). (With a foreword by P. M. Cohn)
No, J.-S., et al.: Binary pseudorandom sequences of period \(2^n -1\) with ideal autocorrelation. IEEE Trans. Inform. Theory 44(2), 814–817 (1998)
Seberry, J., Wysocki, B., Wysocki, T.: On some applications of Hadamard matrices. Metrika 62, 221–239 (2005)
Seberry, J., Yamada, M.: Hadamard matrices, sequences, and block designs. In: Dinitz, J.H., Stinson, D.R. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 431–560. Wiley, New York (1992)
Stern, M.A.: Einige Bemerkungen über eine Determinante. J. Reine Angew. Math. 73, 374–380 (1871)
Stinson, D.R.: Combinatorial Designs, Constructions and Analysis. Springer, New York (2004)
van Lint, J.H., Wilson, R.M.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001)
Wallis, W.D., Street, A.P., Wallis, J.S.: Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices. Lecture Notes in Mathematics, vol. 292. Springer, Heidelberg (1972)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Kotsireas, I.S., Pardalos, P.M. (2014). A New Existence Condition for Hadamard Matrices with Circulant Core. In: Pardalos, P., Resende, M., Vogiatzis, C., Walteros, J. (eds) Learning and Intelligent Optimization. LION 2014. Lecture Notes in Computer Science(), vol 8426. Springer, Cham. https://doi.org/10.1007/978-3-319-09584-4_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-09584-4_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09583-7
Online ISBN: 978-3-319-09584-4
eBook Packages: Computer ScienceComputer Science (R0)