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Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error


The problem of minimization of the decoder error probability is considered for shortened codes of dimension 2m with distance 4 and 6. We prove that shortened Panchenko codes with distance 4 achieve the minimal probability of decoder error under special form of shortening. This shows that Hamming codes are not the best. In the paper, the rules for shortening Panchenko codes are defined and a combinatorial method to minimize the number of words of weight 4 and 5 is developed. There are obtained exact lower bounds on the probability of decoder error and the full solution of the problem of minimization of the decoder error probability for [39,32,4] and [72,64,4] codes. For shortened BCH codes with distance 6, upper and lower bounds on the number of minimal weight codewords are derived. There are constructed [45,32,6] and [79,64,6] BCH codes with the number of weight 6 codewords close to the lower bound and the decoder error probabilities are calculated for these codes. The results are intended for use in memory devices.

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  1. 1.

    E. Fujiwara, Code Design for Dependable Systems Theory and Practical Applications. USA (Wiley, New Jersey, 2006).

    Book  MATH  Google Scholar 

  2. 2.

    R. Micheloni, A. Marelli, and R. Ravasio, Error Correction Codes for Non-Volatile Memories (Springer-Verlag, Qimonda Italy, 2008).

    Google Scholar 

  3. 3.

    Yu. L. Sagalovich, “Code protection of computer random access memory from errors,” Avtom. Telemekh. 52 (5), 3–45 (1991).

    Google Scholar 

  4. 4.

    V. M. Sidel’nikov, “On spectrum of weights of binary Bose–Chaudhuri–Hocquenghem codes,” Probl. Peredachi Inf. 7 (1), 14–22 (1971).

    MathSciNet  Google Scholar 

  5. 5.

    T. Kasami, T. Fujiwara, and S. Lin, “An approximation to the weight destitution of binary linear codes,” IEEE Trans. Inf. Theory 31, 769–780 (1985).

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    I. Krasikov and S. Litsyn, “On Spectra of BCH Codes,” IEEE Trans. Inf. Theory 41, 786–788 (1995).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    E. R. Berlekamp, Algebraic Coding Theory (McGrow-Hill, New-York, 1968; Mir, Moskow, 1971).

    MATH  Google Scholar 

  8. 8.

    T. Kassami, N. Tokura, E. Ivadari, and Ya. Inagaki, Coding Theory (Mir, Moscow, 1978).

    Google Scholar 

  9. 9.

    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland Publ. Company, Amsterdam, 1977).

    MATH  Google Scholar 

  10. 10.

    A. Barg and I. Dumer, “On Computing the Weight Spectrum of Cyclic Codes,” IEEE Trans. Inf. Theory 38, 1382–1386 (1992).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    V. I. Panchenko, “On optimization of linear code with distance 4,” in Proc. 8th All-Union Conf. on Coding Theory and Communications, Kuibyshev, 1981, Part 2: Coding Theory (Moscow, 1981), pp. 132–134 [in Russian].

    Google Scholar 

  12. 12.

    A. A. Davydov and L. M. Tombak, “An alternative to the Hamming code in the class of SEC-DED codes in semiconductor memory,” IEEE Trans. Inf. Theory 37, 897–902 (1991).

    MathSciNet  Article  Google Scholar 

  13. 13.

    R. E. Blahut, Theory and Practice of Error Control Codes (Addison-Wesley, Reading, 1984; Mir, Moscow, 1986).

    MATH  Google Scholar 

  14. 14.

    V. D. Kolesnik, Error-Correcting Coding for Transmission and Storage of Information (Algebraic Theory of Block Codes) (Vysshaya Shkola, Moscow, 2009) [in Russian].

    Google Scholar 

  15. 15.

    A. A. Davydov and L. M. Tombak, “Quasi-perfect linear binary codes with minimal distance 4 and full caps in projective geometry,” Probl. Peredachi Inf. 25 (4), 11–23 (1989).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    A. A. Davydov, A. Yu. Drozhzhina-Labinskaya, and L. M. Tombak, “Supplementary correcting possibilities of BCH codes, correcting double and triple errors,” in Problems of Cybernetics. Complex Engineering of Elemental and Assembly Base of Super Computer, Ed. by V. A. Mel’nikov and Yu. I. Mitropol’skii (VINITI, Moscow, 1988), pp. 86–112 [in Russian].

    Google Scholar 

  17. 17.

    S. A. Ashmanov, Linear Programming (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

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Correspondence to V. B. Afanassiev.

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Original Russian Text © V.B. Afanassiev, A.A. Davydov, D.K. Zigangirov, 2016, published in Informatsionnye Protsessy, 2016, Vol. 16, No. 1, pp. 41–60.

The article was translated by the authors.

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Afanassiev, V.B., Davydov, A.A. & Zigangirov, D.K. Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error. J. Commun. Technol. Electron. 61, 1440–1455 (2016).

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  • binary code
  • probability of decoder error
  • code weight spectrum