Abstract
A code is said to be propelinear if its automorphism group contains a subgroup which acts on the codewords regularly. Such a subgroup is called a propelinear structure on the code. With the aid of computer, we enumerate all propelinear structures on the Nordstrom–Robinson code and analyze the problem of extending them to propelinear structures on the extended Hamming code of length 16. The latter result is based on the description of partitions of the Hamming code of length 16 into Nordstrom–Robinson codes via Fano planes, presented in the paper. As a result, we obtain a record-breaking number of propelinear structures in the class of extended perfect codes of length 16.
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References
Nordstrom, A.W. and Robinson, J.P., An Optimum Nonlinear Code, Inform. Control, 1967, vol. 11, no. 5–6, pp. 613–616.
Semakov, N.V. and Zinoviev, V.A., Complete and Quasi-complete Balanced Codes, Probl. Peredachi Inf., 1969, vol. 5, no. 2, pp. 14–18 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 2, pp. 11–13].
Snover, S.L., The Uniqueness of the Nordstrom–Robinson and Golay Binary Codes, PhD Thesis, Dept. Math., Michigan State Univ., 1973.
Rifà, J., Basart, J.M., and Huguet, L., On Completely Regular Propelinear Codes, Proc. 6th Int. Conf. on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-6), Rome, Italy, July 4–8, 1988, Mora T., Ed., Lect. Notes Comp. Sci., vol. 357, Berlin: Springer, 1989, pp. 341–355.
Borges, J., Mogilnykh, I.Yu., Rifà, J., and Solov’eva, F.I., Structural Properties of Binary Propelinear Codes, Adv. Math. Commun., 2012, vol. 6, no. 3, pp. 329–346.
Forney, G.D., Jr., Trott, M.D., and Sloane, N.J.A., The Nordstrom–Robinson Code is the Binary Image of the Octacode, Coding and Quantization (Proc. DIMACS/IEEE Workshop, Princeton Univ., NJ, USA, Oct. 19–21, 1992), Calderbank, R., Forney, G.D., Jr., and Moayeri, N., Eds., Providence, RI: Amer. Math. Soc., 1993, pp. 19–26.
Preparata, F.P., A Class of Optimum Nonlinear Double-Error-Correcting Codes, Inform. Control, 1968, vol. 13, no. 4, pp. 378–400.
Zaitsev, G.V., Zinoviev, V.A., and Semakov, N.V., Interrelation of Preparata and Hamming Codes and Extension of Hamming Codes to New Double-Error-Correcting Codes, Proc. 2nd Int. Symp. on Information Theory, Tsahkadsor, Armenia, USSR, Sept. 2–8, 1971, Petrov, P.N. and Csaki, F., Eds., Budapest: Akad. Kiadó, 1973, pp. 257–263.
Semakov, N.V. and Zinoviev, V.A., Constant-Weight Codes and Tactical Configurations, Probl. Peredachi Inf., 1969, vol. 5, no. 3, pp. 28–36 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 3, pp. 22–28].
Semakov, N.V., Zinoviev, V.A., and Zaitsev, G.V., Uniformly Packed Codes, Probl. Peredachi Inf., 1971, vol. 7, no. 1, pp. 38–50 [Probl. Inf. Trans. (Engl. Transl.), 1971, vol. 7, no. 1, pp. 30–39].
Hammons, A.R., Jr., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., and Solé, P., The Z4-Linearity of Kerdock, Preparata, Goethals, and Related Codes, IEEE Trans. Inform. Theory, 1994, vol. 40, no. 2, pp. 301–319.
Dumer, I.I., Some New Uniformly Packed Codes, in Proc. Moscow Inst. Physics and Technology, Moscow, 1976, pp. 72–78.
Baker, R.D., van Lint, J.H., and Wilson R.M., On the Preparata and Goethals Codes, IEEE Trans. Inform. Theory, 1983, vol. 29, no. 3, pp. 342–345.
Borges, J., Phelps, K.T., Rifà, J., and Zinoviev, V.A., On Z4-linear Preparata-like and Kerdock-like Codes, IEEE Trans. Inform. Theory, 2003, vol. 49, no. 11, pp. 2834–2843.
Rifà, J. and Pujol, J., Translation-Invariant Propelinear Codes, IEEE Trans. Inform. Theory, 1997, vol. 43, no. 2, pp. 590–598.
Zinoviev, D.V. and Zinoviev, V.A., On the Preparata-like Codes, in Proc. 14th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT-14), Svetlogorsk, Russia, Sept. 7–13, 2014, pp. 342–347.
Solov’eva, F.I. and Topalova, S.T., On Automorphism Groups of Perfect Binary Codes and Steiner Triple Systems, Probl. Peredachi Inf., 2000, vol. 36, no. 4, pp. 53–58 [Probl. Inf. Trans. (Engl. Transl.), 2000, vol. 36, no. 4, pp. 331–335].
Borges, J., Mogilnykh, I.Yu., Rifà, J., and Solov’eva, F.I., On the Number of Nonequivalent Propelinear Extended Perfect Codes, Electron. J. Combin., 2013, vol. 20, no. 2, Research Paper P37.
MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Svyaz’, 1979.
Bierbrauer, J., Nordstrom–Robinson Code and A7-Geometry, Finite Fields Appl., 2007, vol. 13, no. 1, pp. 158–170.
Berlekamp, E.R., Coding Theory and the Mathieu Groups, Inform. Control, 1971, vol. 18, no. 1, pp. 40–64.
Avgustinovich, S.V. and Solov’eva, F.I., To the Metric Rigidity of Binary Codes, Probl. Peredachi Inf., 2003, vol. 39, no. 2, pp. 23–28 [Probl. Inf. Trans. (Engl. Transl.), 2003, vol. 39, no. 2, pp. 178–183].
Mogilnykh, I.Yu., On Weak Isometries of Preparata Codes, Probl. Peredachi Inf., 2009, vol. 45, no. 2, pp. 78–83 [Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 2, pp. 145–150].
Fernández-Córdoba, C. and Phelps, K.T., On the Minimum Distance Graph of an Extended Preparata Code, Des. Codes Cryptogr., 2010, vol. 57, no. 2, pp. 161–168.
Avgustinovich, S.V., To the Structure of Minimum Distance Graphs of Perfect Binary (n, 3)-Codes, Diskretn. Anal. Issled. Oper., Ser. 1, 1998, vol. 5, no. 4, pp. 3–5.
Mogilnykh, I.Yu., Östergård, P.R.J., Pottonen, O., and Solov’eva, F.I., Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs, IEEE Trans. Inform. Theory, 2009, vol. 55, no. 6, pp. 2622–2625.
McEliece, R.J., A Public-Key Cryptosystem Based on Algebraic Coding Theory, JPL DSN Progress Report, 1978, pp. 114–116.
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Original Russian Text © I.Yu. Mogil’nykh, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 3, pp. 97–107.
The results of Section 3 of the paper are obtained under the support of the Russian Foundation for Basic Research, project no. 13-01-00463; results of Section 4 are funded by the Russian Science Foundation, project no. 14-11-00555.
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Mogil’nykh, I.Y. On extending propelinear structures of the Nordstrom–Robinson code to the Hamming code. Probl Inf Transm 52, 289–298 (2016). https://doi.org/10.1134/S0032946016030078
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DOI: https://doi.org/10.1134/S0032946016030078