Skip to main content

An inductive construction of minimal codes


We provide new families of minimal codes in any characteristic, useful for the construction of secret sharing schemes. Also, an inductive construction of minimal codes is presented.

This is a preview of subscription content, access via your institution.


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

    Article  MathSciNet  Google Scholar 

  2. Bartoli, D, Bonini, M: Minimal linear codes in odd characteristic. IEEE Trans. Inform. Theory. 65, 4152–4155 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. of AFIPS National Computer Conference, pp. 313–317. New York (1979)

  5. Bonini, M., Borello, M.: Minimal linear codes arising from blocking sets. J. Algebr. Comb. (2020)

  6. Carlet, C., Ding, C., Yuan, J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inf. Theory. 51, 2089–2102 (2005)

    Article  Google Scholar 

  7. Chabanne, H., Cohen, G., Patey, A.: Towards secure two-party computation from the wire-tap channel. In: Information Security and Cryptology – ICISC 2013, pp. 34–46. Heidelberg (2014)

  8. Chang, S., Hyun, J.Y.: Linear codes from simplicial complexes. Des. Codes Cryptogr. 86, 2167–2181 (2018)

    Article  MathSciNet  Google Scholar 

  9. Bruck, J, Naor, M.: The hardness of decoding linear codes with preprocessing. IEEE Trans. Inform. Theory. 36, 381–385 (1990)

    Article  MathSciNet  Google Scholar 

  10. Cohen, G.D., Mesnager, S., Patey, A.: On minimal and quasi-minimal linear codes. In: IMACC 2013 pp. 85–98. Heidelberg (2013)

  11. Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory. 60, 3265–3275 (2015)

    Article  MathSciNet  Google Scholar 

  12. Ding, C., Heng, Z., Zhou, Z.: Minimal binary linear codes. IEEE Trans. Inf. Theory. 64, 6536–6545 (2018)

    Article  MathSciNet  Google Scholar 

  13. Ding, C., Li, N., Li, C., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discrete Math. 339, 415–427 (2016)

    Article  MathSciNet  Google Scholar 

  14. Ding, C., Yuan, J.: Covering and secret sharing with linear codes. In: DMTCS 2003, pp.11–25. Heidelberg (2003)

  15. Heng, Z., Ding, C., Zhou, Z: Minimal linear codes over finite fields. Finite Fields Appl. 54, 176–196 (2018)

    Article  MathSciNet  Google Scholar 

  16. Klove, T.: Codes for Error Detection Series on Coding Theory and Cryptology, vol. 2. World Scientific Publishing Co. Pte. Ltd., Hackensack (2007)

    Google Scholar 

  17. Massey, J.L.: Minimal codewords and secret sharing (1993)

  18. Massey, J.L.: Some applications of coding theory in cryptography. In Codes and Cyphers: Cryptography and Coding IV, pp. 33–47. Esses (1995)

  19. Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)

    Article  MathSciNet  Google Scholar 

  20. Song, Y., Li, Z.: Secret sharing with a class of minimal linear codes. arXiv: (2012). Accessed 20 October 2020

  21. Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory. 52, 206–212 (2006)

    Article  MathSciNet  Google Scholar 

Download references


The research of D. Bartoli was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).

The research of M. Bonini was supported by the Irish Research Council, grant n. GOIPD/2020/597.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Matteo Bonini.

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

Bartoli, D., Bonini, M. & Güneş, B. An inductive construction of minimal codes. Cryptogr. Commun. 13, 439–449 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification (2010)