Skip to main content
SpringerLink
Log in

Characteristic vector and weight distribution of a linear code

  • Original Research
  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

An algorithm for computing the weight distribution of a linear [n,k] code over a finite field \(\mathbb {F}_{q}\) is developed. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code \(\mathcal {S}_{q,k}\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baart, R., Boothby, T., Cramwinckel, J., Fields, J., Joyner, D., Miller, R., Minkes, E., Roijackers, E., Ruscio, L., Tjhai, C.: GAP package GUAVA. https://www.gap-system.org/Packages/guava.html

  2. Bellini, E., Sala, M.: A deterministic algorithm for the distance and weight distribution of binary nonlinear codes. Int. J. Inform. Coding Theory (IJICOT) 5, 18–35 (2018)

    Article  MathSciNet  Google Scholar 

  3. Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.: On the inherent intractability of certain coding problems. IEEE Trans. Inform. Theory 24, 384–386 (1978)

    Article  MathSciNet  Google Scholar 

  4. Bikov, D., Bouyukliev, I.: Parallel fast walsh transform algorithm and its implementation with CUDA on GPUs. Cybern. Inform. Technol. 18, 21–43 (2018)

    MathSciNet  Google Scholar 

  5. Borges-Quintana, M., Borges-Trenard, M.A., Fitzpatrick, P., Martínez-Moro, E.: Groebner bases and combinatorics for binary codes. Appl. Algebra Eng. Comm. Comput. 19, 393–411 (2008)

    Article  Google Scholar 

  6. Borges-Quintana, M., Borges-Trenard, M.A., Márquez-Corbella, I., Martínez-Moro, E.: Computing coset leaders and leader codewords of binary codes. J. Algebra Appl. 14(8), 1550128 (2015)

    Article  MathSciNet  Google Scholar 

  7. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system i: The user language. J. Symbolic Comput. 24, 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125

    Article  MathSciNet  Google Scholar 

  8. Bouyukliev, I.: What is Q-Extension? Serdica. J. Comput. 1, 115–130 (2007)

    MathSciNet  MATH  Google Scholar 

  9. Bouyukliev, I., Bakoev, V.: A method for efficiently computing the number of codewords of fixed weights in linear codes. Discret. Appl. Math. 156, 2986–3004 (2008)

    Article  MathSciNet  Google Scholar 

  10. Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, in: Encyclopedia Math. Appl., vol. 134. Cambridge University Press (2010)

  11. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Crypt. 15(2), 125–156 (1998)

    Article  MathSciNet  Google Scholar 

  12. Dodunekov, S., Simonis, J.: Codes and projective multisets. Electron. J. Comb. 5(1), 37 (1998)

    Article  MathSciNet  Google Scholar 

  13. Dodunekova, R., Dodunekov, S.M.: Sufficient conditions for good and proper error-detecting codes. IEEE Trans. Inform. Theory 43(6), 2023–2026 (1997)

    Article  MathSciNet  Google Scholar 

  14. Edel, Y., Pott, A.: On the equivalence of nonlinear functions. In: Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, NATO Science for Peace and Security Series - D: Information and Communication Security, vol. 23, pp 87–103. IOS Press (2009)

  15. Elliott, D.F., Rao, K.R.: Fast Transforms: Algorithms, Analises, Applications. Academic Press, Orlando (1982)

    Google Scholar 

  16. Giorgetti, M., Sala, M.: A commutative algebra approach to linear codes. J. Algebra 321(8), 2259–2286 (2009)

    Article  MathSciNet  Google Scholar 

  17. Good, I.J.: The interaction algorithm and practical fourier analysis. J. Royal Stat. Soc. 20(2), 361–372 (1958)

    MathSciNet  MATH  Google Scholar 

  18. Gulliver, T., Bhargava, V., Stein, J.: Q-ary Gray codes and weight distributions. Appl. Math Comput. 103, 97–109 (1999)

    MathSciNet  MATH  Google Scholar 

  19. Han, S., Seo, H.S., Ju, S.: Efficient calculation of the weight distributions for linear codes over large finite fields. Contemporary Engineering Sciences 9, 609–617 (2016)

    Article  Google Scholar 

  20. Huffman, W.C., Pless, V.: Fundamentals of error-correcting codes. Cambridge Univ. Press (2003)

  21. Jan, B., Montrucchio, B., Ragusa, C., Khan, F.G., Khan, O.: Parallel butterfly sorting algorithm on GPU. In: Proceedings of Artificial Intelligence and Applications, pp 795–026, Innsbruck, Austria (2013)

  22. Joux, A.: Algorithmic Cryptanalysis. Chapman & Hall/CRC, Boca Raton (2009)

    Book  Google Scholar 

  23. Jurrius, R., Pellikaan, R.: Codes, arrangements and matroids. In: Algebraic geometry modeling in information theory, in: Series on Coding Theory and Cryptology, vol. 8. World Scientific Publishing (2013)

  24. Karpovsky, M.G.: On the weight distribution of binary linear codes. IEEE Trans. Inform. Theory 25(1), 105–109 (1979)

    Article  MathSciNet  Google Scholar 

  25. Katsman, G.L., Tsfasman, M.A.: Spectra of algebraic-geometric codes. Probl. Peredachi Inf. 23(4), 19–34 (1987)

    MATH  Google Scholar 

  26. MacWilliams, F., Mallows, C., Sloane, N.: Generalizations of Gleason’s theorem on weight enumerators of self-dual codes. IEEE Trans. Inform. Theory 18 (6), 794–805 (1972)

    Article  MathSciNet  Google Scholar 

  27. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier, North-Holland (1977)

    MATH  Google Scholar 

  28. Márquez-Corbella, I., Martínez-Moro, E., Suárez-Canedo, E.: On the ideal associated to a linear code. Advances in Mathematics of Communications 10(2), 229–254 (2016)

    Article  MathSciNet  Google Scholar 

  29. Parhami, B.: Introduction to Parallel Processing: Algorithms and Architectures, Series in Computer Science. Springer, Berlin (2002)

    Google Scholar 

  30. Sala, M.: Gröbner basis techniques to compute weight distributions of shortened cyclic codes. J. Algebra Appl. 6, 403–414 (2007)

    Article  MathSciNet  Google Scholar 

  31. Sendrier, N.: Finding the permutation between equivalent linear codes: The support splitting algorithm. IEEE Trans. Inform. Theory 46, 1193–1203 (2000)

    Article  MathSciNet  Google Scholar 

  32. Stichtenoth, H.: Subfield subcodes and trace codes. In: Algebraic function fields and codes. graduate texts in mathematics, vol. 254. Springer, Berlin (2009)

  33. Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Trans. Inform. Theory 43(6), 1757–1766 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported by Grants DN 02/2/13.12.2016 and KP-06-N32/2-2019 of the Bulgarian National Science Fund.

We thank Geoff Bailey, Computational Algebra Group, University of Sydney, for the provided information about the processor used by Magma Calculator. We are greatly indebted to the unknown referees for their useful suggestions.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, P.O. Box 323, Veliko Tarnovo, Bulgaria

    Iliya Bouyukliev & Paskal Piperkov

  2. Faculty of Mathematics and Informatics, St. Cyril and St. Methodius University of Veliko Tarnovo, Veliko Tarnovo, Bulgaria

    Stefka Bouyuklieva

  3. Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka, 599-8531, Japan

    Tatsuya Maruta

Authors
  1. Iliya Bouyukliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefka Bouyuklieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tatsuya Maruta
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paskal Piperkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefka Bouyuklieva.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouyukliev, I., Bouyuklieva, S., Maruta, T. et al. Characteristic vector and weight distribution of a linear code. Cryptogr. Commun. 13, 263–282 (2021). https://doi.org/10.1007/s12095-020-00458-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-020-00458-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation