Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
125 tokens/sec
GPT-4o
53 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Minimal Ternary Linear Codes from Vectorial Functions (2403.11775v1)

Published 18 Mar 2024 in cs.IT and math.IT

Abstract: The study on minimal linear codes has received great attention due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, to the best of our knowledge, no infinite family of minimal ternary linear codes was found from vectorial functions. In this paper, we present a necessary and sufficient condition for a large class of ternary linear codes from vectorial functions such that those codes are minimal. Based on that, we construct several minimal ternary linear codes with three-weight from vectorial regular plateaued functions, and determine their weight distributions. Moreover, we also give a necessary and sufficient condition for a large family of ternary linear codes from vectorial functions such that the codes are minimal and violate the AB condition simultaneously. According to this characterization, we find several minimal ternary linear codes violating the AB condition. Notably, our results show that our method can be applied to solve a problem on minimal linear codes proposed by Li et al.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (41)
  1. A. Ashikhmin and A. Barg, “Minimal vectors in linear codes,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 2010-2017, Sep. 1998.
  2. D. Bartoli and M. Bonini, “Minimal linear codes in odd characteristic,” IEEE Trans. Inf. Theory, vol. 65, no. 7, pp. 4152-4155, Jul. 2019.
  3. M. Bonini and M. Borello, “Minimal linear codes arising from blocking sets,” J. Algebr. Combinatorics, vol. 53, pp. 327-341, 2021.
  4. C. Carlet, C. Ding, and J. Yuan, “Linear codes from perfect nonlinear mappings and their secret sharing schemes,” IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 2089-2102, Jun. 2005.
  5. H. Chabanne, G. Cohen, and A. Patey, “Towards secure two-party computation from the wire-tap channel,” in Proc. ICISC, in Lecture Notes in Computer Science, vol. 8565, H.-S. Lee and D.-G. Han, Eds. Berlin, Germany: Springer-Verlag, 2014, pp. 34-46.
  6. S. Chang and J. Y. Hyun, “Linear codes from simplicial complexes,” Des., Codes Cryptogr., vol. 86, no. 10, pp. 2167-2181, Oct. 2018.
  7. G. D. Cohen, S. Mesnager, and A. Patey, “On minimal and quasiminimal linear codes,” in Proc. IMA Int. Conf. Cryptogr. Coding, M. Stam, Eds. Berlin, Germany: Springer-Verlag, 2013, pp. 85-98.
  8. C. Ding, “Linear codes from some 2-designs,” IEEE Trans. Inf. Theory, vol. 60, no. 6, pp. 3265-3275, Jun. 2015.
  9. C. Ding, Y. Gao, and Z. Zhou, “Five families of three-Weight ternary cyclic codes and their duals,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 7940-7946, 2013.
  10. C. Ding, Z. Heng, and Z. Zhou, “Minimal binary linear codes,” IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, Oct. 2018.
  11. C. Ding and J. Yuan, “Covering and secret sharing with linear codes,” in Discrete Mathematics and Theoretical Computer Science, vol. 2731. Berlin, Germany: Springer-Verlag, 2003, pp. 11-25.
  12. K. Ding and C. Ding, “A class of two-weight and three-weight codes and their applications in secret sharing,” IEEE Trans. Inf. Theory, vol. 61, no. 11, pp. 5835-5842, Nov. 2015.
  13. X. Du, R. Rodríguez, and H. Wu, “Infinite families of minimal binary codes via Krawtchouk polynomials”, Des. Codes Cryptogr., 2024. https://doi.org/10.1007/s10623-023-01353-y
  14. Z. Heng, C. Ding, and Z. Zhou, “Minimal linear codes over finite fields,” Finite Fields Appl., vol. 54, pp. 176-196, Nov. 2018.
  15. Z. Heng and Q. Yue, “Two classes of two-weight linear codes,” Finite Fields Appl., vol. 38, pp. 72-92, 2016.
  16. J. Y. Hyun, J. Lee, and Y. Lee, “Explicit criteria for construction of plateaued functions,” IEEE Trans. Inf. Theory, vol. 62, no. 12, pp. 7555-7565, 2016.
  17. Y. Li, H. Kan, S. Mesnager, J. Peng, C.H. Tan, and L. Zheng, “Generic constructions of (Boolean and vectorial) bent functions and their consequences,” IEEE Trans. Inf. Theory, vol. 68, no. 4, pp. 2735-2751, 2022.
  18. Y. Li, J. Peng, H. Kan, and L. Zheng, “Minimal binary linear codes from vectorial Boolean functions,” IEEE Trans. Inf. Theory, vol. 69, no. 5, pp. 2955-2968, 2023.
  19. G. Luo and X. Cao, “Five classes of optimal two-weight linear codes,” Cryptogr. Commun., vol. 10, no. 6, pp. 1119-1135, 2018.
  20. S. Mesnager, “Linear codes with few weights from weakly regular bent functions based on a generic construction,” Cryptogr. Commun., vol. 9, no. 1, pp. 71-84, Jan. 2017.
  21. S. Mesnager, F. Özbudak, and A. Sınak, “Linear codes from weakly regular plateaued functions and their secret sharing schemes,” Des., Codes Cryptogr., vol. 87, pp. 463-480, 2019.
  22. S. Mesnager, Y. Qi, H. Ru, and C. Tang, “Minimal linear codes from characteristic functions,” IEEE Trans. Inf. Theory, vol. 66, no. 9, pp. 5404-5413, Sep. 2020.
  23. S. Mesnager, L. Qian, X. Cao, and Mu Yuan, “Several families of binary minimal linear codes from two-to-one functions,” IEEE Trans. Inf. Theory, vol. 69, no. 5, pp. 3285-3301, 2023.
  24. S. Mesnager and A. Sınak, “Several classes of minimal linear codes with few weights from weakly regular plateaued functions,” IEEE Trans. Inf. Theory, vol. 66, no. 4, pp. 2296-2310, 2020.
  25. K. Nyberg, “Perfect nonlinear S-boxes,” in Advances in Cryptology-EUROCRYPT (Lecture Notes in Computer Science), vol. 547. Berlin, Germany: Springer-Verlag, 1991, pp. 378-385.
  26. E. Pasalic, R. Rodríguez, F. Zhang, and Y. Wei, “Several classes of minimal binary linear codes violating the Aschikhmin-Barg bound,” Cryptogr. Commun., vol. 13, pp. 637-659, 2021.
  27. R. M. Pelen. “Three weight ternary linear codes from non-weakly regular bent functions,” Advances in Mathematics of Communications, 2022, doi: 10.3934/amc.2022020.
  28. R. Rodríguez, E. Pasalic, F. Zhang, and Y. Wei, “Minimal p𝑝pitalic_p-ary codes via the direct sum of functions, non-covering permutations and subspaces of derivatives,” IEEE Trans. Inf. Theory, 2023, doi: 10.1109/TIT.2023.3338577.
  29. M. Shi, Y. Guan, and P. Solé, “Two new families of two-weight codes,” IEEE Trans. Inf. Theory, vol. 63, no. 10, pp. 6240-6246, 2017.
  30. C. Tang, N. Li, Y. Qi, Z. Zhou, and T. Helleseth, “Linear codes with two or three weights from weakly regular bent functions,” IEEE Trans. Inf. Theory, vol. 62, no. 3, pp. 1166-1176, Mar. 2016.
  31. R. Tao, T. Feng, and W. Li, “A construction of minimal linear codes from partial difference sets,” IEEE Trans. Inf. Theory, vol. 67, no. 6, pp. 3724-3734, 2021.
  32. Y. Xia and C. Li, “Three-weight ternary linear codes from a family of power functions,” Finite Fields Appl., vol. 46, pp. 17-37, 2017.
  33. G. Xu and L. Qu, “Three classes of minimal linear codes over the finite fields of odd characteristic,” IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019.
  34. G. Xu, L. Qu and X. Cao, “Minimal linear codes from Maiorana-McFarland functions,” Finite Fields Appl., vol. 65, 2020, Art. no. 101688.
  35. G. Xu, L. Qu, and G. Luo, “Minimal linear codes from weakly regular bent functions,” Cryptogr. Commun., vol. 14, pp. 415-431, 2022.
  36. S. Yang and Z. Yao, “Complete weight enumerators of a family of three-weight linear codes,” Des. Codes Cryptogr., vol. 82, pp. 663-674, 2017.
  37. J. Yuan and C. Ding,“Secret sharing schemes from three classes of linear codes,” IEEE Trans. Inf. Theory, vol. 52, no. 1, pp. 206-212, Jan. 2006.
  38. F. Zhang, E. Pasalic, R. Rodríguez, and Y. Wei, “Wide minimal binary linear codes from the general Maiorana-McFarland class,” Des. Codes Cryptogr., vol. 89, pp. 1485-1507, 2021.
  39. F. Zhang, E. Pasalic, R. Rodríguez, and Y. Wei, “Minimal binary linear codes: a general framework based on bent concatenation,” Des. Codes Cryptogr., vol. 90, pp. 1289-1318, 2022.
  40. Z. Zhou, “Three-weight ternary linear codes from a family of cyclic difference sets,” Des. Codes Cryptogr., vol. 86, pp. 2513-2523, 2018.
  41. Z. Zhou, N. Li, C. Fan, and T. Helleseth, “Linear codes with two or three weights from quadratic bent functions,” Des. Codes Cryptogr., vol. 81, no. 2, pp. 283-295, 2016.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now