Abstract
We study \(1\)-perfect codes in Doob graphs \(D(m,n)\). We show that such codes that are linear over the Galois ring \(\mathrm {GR}(4^2)\) exist if and only if there exist integers \(\gamma \ge 0\) and \(\delta >0\) such that \(n=(4^{\gamma +\delta }-1)/3\) and \(m=(4^{\gamma +2\delta }-4^{\gamma +\delta })/6\). We also prove necessary conditions on \((m,n)\) for \(1\)-perfect codes that are linear over \(Z_4\) (we call such codes additive) to exist in \(D(m,n)\) graphs; for some of these parameters, we show the existence of codes. For every \(m\) and \(n\) satisfying \(2m+n = (4^\mu - 1)/3\) and \(m\le (4^\mu -5\cdot 2^{\mu -1}+1)/9\), we construct \(1\)-perfect codes in \(D(m,n)\), which do not necessarily have a group structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borges J., Rifà J.: A characterization of \(1\)-perfect additive codes. IEEE Trans. Inf. Theory 45(5), 1688–1697 (1999). doi:10.1109/18.771247.
Chihara L.: On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18(1), 191–207 (1987). doi:10.1137/0518015.
Cvetković D.M., Doob M., Sachs H.: Spectra of Graphs: Theory and Application. Academic Press, New York (1980).
Etzion T.: Configuration distribution and designs of codes in the Johnson scheme. J. Comb. Des. 15(1), 15–34 (2007). doi:10.1002/jcd.20102.
Golay M.J.E.: Notes on digital coding. Proc. IRE 37(6), 657 (1949). doi:10.1109/JRPROC.1949.233620.
Gordon D.M.: Perfect single error-correcting codes in the Johnson scheme. IEEE Trans. Inf. Theory 52(10), 4670–4672 (2006). doi:10.1109/TIT.2006.881744.
Güzeltepe M., Heden O.: Perfect Mannheim, Lipschitz and Hurwitz weight codes. Math. Commun. 19(2), 253–276 (2014).
Hamming R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 29(2), 147–160 (1950).
Heden O.: On perfect codes over non prime power alphabets. In: Bruen A.A., Wehlau D.L. (eds.) Error-Correcting Codes, Finite Geometries and Cryptography, vol. 523, pp. 173–184. Contemporary Mathematics/AMS, Providence (2010). doi:http://dx.doi.org/10.1090/conm/523.
Heden O., Güzeltepe M.: On perfect 1 \(\cal E\)-error-correcting codes. Math. Commun. 20(1), to appear (2015).
Koolen J.H., Munemasa A.: Tight \(2\)-designs and perfect \(1\)-codes in Doob graphs. J. Stat. Plann. Inference 86(2), 505–513 (2000). doi:10.1016/S0378-3758(99)00126-3.
Martin W.J., Zhu X.J.: Anticodes for the Grassman and bilinear forms graphs. Des. Codes Cryptogr. 6(1), 73–79 (1995). doi:10.1007/BF01390772.
Mollard M.: A generalized parity function and its use in the construction of perfect codes. SIAM J. Algebr. Discret. Methods 7(1), 113–115 (1986). doi:10.1137/0607013.
Phelps K.T.: A product construction for perfect codes over arbitrary alphabets. IEEE Trans. Inf. Theory 30(5), 769–771 (1984). doi:10.1109/TIT.1984.1056963.
Tietäväinen A.: On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24(1), 88–96 (1973). doi:10.1137/0124010.
Zinoviev V., Leontiev V.: The nonexistence of perfect codes over Galois fields. Probl. Control Inf. Theory 2(2), 123–132, 16–24 [Engl. transl.] (1973).
Acknowledgments
The work was funded by the Russian Science Foundation (Grant 14-11-00555).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Etzion.
Rights and permissions
About this article
Cite this article
Krotov, D.S. Perfect codes in Doob graphs. Des. Codes Cryptogr. 80, 91–102 (2016). https://doi.org/10.1007/s10623-015-0066-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-015-0066-6