Estimating the fraction of erasure patterns correctable by linear codes


The conditional probability (fraction) of the successful decoding of erasure patterns of high (greater than the code distance) weights is investigated for linear codes with the partially known or unknown weight spectra of code words. The estimated conditional probabilities and the methods used to calculate them refer to arbitrary binary linear codes and binary Hamming, Panchenko, and Bose–Chaudhuri–Hocquenghem (BCH) codes, including their extended and shortened forms. Error detection probabilities are estimated under erasure-correction conditions. The product-code decoding algorithms involving the correction of high weight erasures by means of component Hamming, Panchenko, and BCH codes are proposed, and the upper estimate of decoding failure probability is presented.

This is a preview of subscription content, log in to check access.


  1. 1.

    G. D. Forney, “Exponential error bounds for erasure, list, and decision feedback schemes,” IEEE Trans. Inf. Theory 14, 206–220 (1968).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    V. V. Zyablov and P. S. Rybin, “Erasure correction by low-density codes,” Probl. Inf. Transm. 45 (3), 204–220 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    M. N. Nazarov and S. P. Mishin, “Almost optimum codes used to correct erasures,” in Vestn. Volgograd. Gos. Univ., Ser. 1: Mat., Fiz., No. 4, 59–69 (1999) [in Russian].

    Google Scholar 

  4. 4.

    A. Ashikhmin and A. Barg, “Minimal vectors in linear codes,” IEEE Trans. Inf. Theory 44, 2010–2017 (1998).

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    O. V. Popov, “On assessment of linear code ability to correct erasures and find errors in the presence of erasures,” Elektrosvyaz’, No. 10, 1 (1967) [in Russian].

    Google Scholar 

  6. 6.

    O. V. Popov, On Correction of Erasures by Cyclic Codes. Digital Data Transmission over Channels with Memory (Nauka, Moscow, 1970), pp. 111–124 [in Russian].

    Google Scholar 

  7. 7.

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

    Article  Google Scholar 

  8. 8.

    M. Bossert, M. Braitbakh, V. V. Zyablov, and V. P. Sidorenko, “Codes correcting the set of error spots or erasures,” Probl. Inf. Transmis. 33 (4), 297–306 (1997).

    MATH  Google Scholar 

  9. 9.

    A. A. Davydov, A. Yu. Drozhzhina-Labinskaya, and L. M. Tombak, “Supplementary correcting possibilities of BCH codes correcting double errors and finding triple errors,” in Cybernetics questions. Complex design of element and design base of the supercomputer, by Ed. V. A. Mel’nikov, and Yu. I. Mitropol’skii (VINITI, Moscow, 1988), pp. 86–112 [in Russian].

    Google Scholar 

  10. 10.

    A. A. Davydov, L. P. Kaplan, Yu. V. Smerkis, and G. L. Tauglikh, “Optimization of shortened Hamming codes,” Probl. Inf. Transmis. 17 (4), 261–267 (1981).

    MathSciNet  MATH  Google Scholar 

  11. 11.

    A. A. Davydov and L. M. Tombak, “Number of words with minimum weights in block codes,” Probl. Inf. Transmis. 24 (1), 11–24 (1988).

    MATH  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.

    V. I. Panchenko, “On optimization of a linear code with distance 4,” in Proc. VIII All-Union Conf. on According to the Theory of Coding and Information Transfer, Kuibyshev, 1981, Part 2: The Theory of Coding (Akad. Nauk SSSR, Moscow, 1981), pp. 132–134 [in Russian].

    Google Scholar 

  14. 14.

    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 

  15. 15.

    K. M. Cheung, “The weight distribution and randomness of linear codes,” in TDA Progress Report 42-97, Jet Propulsion Lab., Pasadena, CA, California, USA, 1989 (Inst. of Tech., Pasadena, 1989), pp. 208–215. Accessed 22.11.2016.

    Google Scholar 

  16. 16.

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

    Google Scholar 

  17. 17.

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

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    F. J. MacWilliams and N. J. A. Sloane, The Theory Error-Correcting Codes (North-Holland, Amsterdam, 1977; Svyaz’, Moscow, 1979).

    Google Scholar 

  19. 19.

    R. H. Morelos-Zaragoza, The Art of Error Correcting Coding (Wiley, Chichester, 2002; Tekhnosfera, Moscow, 2005).

    Google Scholar 

  20. 20.

    V. M. Sidel’nikov, “On spectrum of weights of binary Bose–Chaudhuri–Hocquenghem codes,” Probl. Inf. Transmis. 7 (1), 11–17 (1971).

    MathSciNet  Google Scholar 

  21. 21.

    Weight Distribution. (Accessed 27.11.2016).

Download references

Author information



Corresponding author

Correspondence to V. B. Afanassiev.

Additional information

Original Russian Text © V.B. Afanassiev, A.A. Davydov, D.K. Zigangirov, 2016, published in Informatsionnye Protsessy, 2016, Vol. 16, No. 4, pp. 382–404.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Afanassiev, V.B., Davydov, A.A. & Zigangirov, D.K. Estimating the fraction of erasure patterns correctable by linear codes. J. Commun. Technol. Electron. 62, 669–685 (2017).

Download citation


  • linear code
  • erasure correction
  • product code