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On new completely regular q-ary codes

  • Coding Theory
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Abstract

In this paper, new completely regular q-ary codes are constructed from q-ary perfect codes. In particular, several new ternary completely regular codes are obtained from the ternary [11, 6, 5] Golay code. One of these codes with parameters [11, 5, 6] has covering radius ρ = 5 and intersection array (22, 20, 18, 2, 1; 1, 2, 9, 20, 22). This code is dual to the ternary perfect [11, 6, 5] Golay code. Another [10, 5, 5] code has covering radius ρ = 4 and intersection array (20, 18, 4, 1; 1, 2, 18, 20). This code is obtained by deleting one position of the former code. All together, the ternary Golay code results in eight completely regular codes, only four of which were previously known. Also, new infinite families of completely regular codes are constructed from q-ary Hamming codes.

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References

  1. Delsarte, P., An Algebraic Approach to the Association Schemes of Coding Theory, Philips Res. Rep. Suppl., 1973, no. 10.

  2. Neumaier, A., Completely Regular Codes, Discrete Math., 1992, vol. 106/107, pp. 353–360.

    Article  Google Scholar 

  3. Tietäväinen, A., On the Nonexistence of Perfect Codes over Finite Fields, SIAM J. Appl. Math., 1973, vol. 24, no. 1, pp. 88–96.

    Article  Google Scholar 

  4. Zinoviev, V.A. and Leontiev, V.K., The Nonexistence of Perfect Codes over Galois Fields, Probl. Control Inf. Theory, 1973, vol. 2, no. 2, pp. 16–24.

    Google Scholar 

  5. van Tilborg, H.C.A., Uniformly Packed Codes, PhD Thesis, Tech. Univ. Eindhoven, 1976.

  6. 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].

    Google Scholar 

  7. Goethals, J.-M. and van Tilborg, H.C.A., Uniformly Packed Codes, Philips Res. Rep., 1975, vol. 30, pp. 9–36.

    MATH  Google Scholar 

  8. Borges, J., Rifà, J., and Zinoviev, V.A., On New Completely Regular Binary Codes, in Proc. 4th Int. Workshop on Optimal Codes and Related Topics, Pamporovo, Bulgaria, 2005, pp. 32–35.

  9. Borges, J., Rifà, J., and Zinoviev, V.A., On Non-antipodal Binary Completely Regular Codes, accepted for publication in Discrete Math.

  10. Borges, J., Rifà, J., and Zinoviev, V.A., On Non-antipodal Binary Completely Regular Codes, in Proc. 10th Int. Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, Russia, 2006, pp. 35–39.

  11. Rifà, J. and Zinoviev, V.A., On Completely Regular Codes from Perfect Codes, in Proc. 10th Int. Workshop on Algebraic and Combinatorial Coding Theory, Zvenigorod, Russia, 2006, pp. 225–229.

  12. Bassalygo, L.A., Zaitsev, G.V., and Zinoviev, V.A., Uniformly Packed Codes, Probl. Peredachi Inf., 1974, vol. 10, no. 1, pp. 9–14 [Probl. Inf. Trans. (Engl. Transl.), 1974, vol. 10, no. 1, pp. 6–9].

    Google Scholar 

  13. Bassalygo, L.A. and Zinoviev, V.A., Remark on Uniformly Packed Codes, Probl. Peredachi Inf., 1977, vol. 13, no. 3, pp. 22–25 [Probl. Inf. Trans. (Engl. Transl.), 1977, vol. 13, no. 3, pp. 178–180].

    Google Scholar 

  14. Zinoviev, V.A. and Leontiev, V.K., On Perfect Codes, Probl. Peredachi Inf., 1972, vol. 8, no. 1, pp. 26–35 [Probl. Inf. Trans. (Engl. Transl.), 1972, vol. 8, no. 1, pp. 17–24].

    Google Scholar 

  15. Semakov, N.V. and Zinoviev, V.A., Constant-Weight Codes and Tactical Configurations, Probl. Peredachi Inf., 1969, vol. 5, no. 3, pp. 29–38 [Probl. Inf. Trans. (Engl. Transl.), 1969, vol. 5, no. 3, pp. 22–28].

    Google Scholar 

  16. Brouwer, A.E., Cohen, A.M., and Neumaier, A., Distance-Regular Graphs, Berlin: Springer, 1989.

    MATH  Google Scholar 

  17. 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.

    MATH  Google Scholar 

  18. Bassalygo, L.A., Zinoviev, V.A., Leont’ev, V.K., and Fel’dman, N.I., Nonexistence of Perfect Codes for Some Composite Alphabets, Probl. Peredachi Inf., 1975, vol. 11, no. 3, pp. 3–13 [Probl. Inf. Trans. (Engl. Transl.), 1975, vol. 11, no. 3, pp. 181–189].

    Google Scholar 

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Original Russian Text © V.A. Zinoviev, J. Rifà, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 2, pp. 34–51.

Supported in part by the CICYT, Grants TIC2003-08604-C04-01 and TIC2003-02041, the Catalan DURSI, Grants 2001SGR 00219 and 2004PIV1-3, and the Russian Foundation for Basic Research, project nos. 03-01-00098 and 06-01-00226.

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Zinoviev, V.A., Rifà, J. On new completely regular q-ary codes. Probl Inf Transm 43, 97–112 (2007). https://doi.org/10.1134/S0032946007020032

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