doi: 10.3934/amc.2022027
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Some constructions of (almost) optimally extendable linear codes

1. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

2. 

School of Cyberspace, Hangzhou Dianzi University, Hangzhou, 310018, China

* Corresponding author: Qin Yue

Received  December 2021 Early access April 2022

Fund Project: This paper was supported by National Natural Science Foundation of China (Nos. 61772015, 62172219)

Let $ G $ be a generator matrix of a linear code $ \mathcal C $ and $ [G: I_k] $ be a generator matrix of its extendable linear code $ \mathcal {C}' $, we call $ \mathcal C $ is optimally (almost optimally) extendable if $ d(\mathcal C^\perp) = d({\mathcal C'}^\perp) $($ d(\mathcal C^\perp) $ is very close to $ d({\mathcal C'}^\perp) $, respectively), where $ d(\mathcal C^\perp) $ is the minimal distance of the dual code of $ \mathcal C $. In order to safeguard the susceptible information lay in registers oppose SCA and FIA, it is useful to construct an optimally extendable linear code $ \mathcal C $. In this paper, we construct three classes of (almost) optimally extendable linear codes: (1) irreducible cyclic codes; (2) maximum-distance-separable (MDS) codes and near maximum-distance-separable (NMDS) codes.

Citation: Xiaoshan Quan, Qin Yue, Liqin Hu. Some constructions of (almost) optimally extendable linear codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022027
References:
[1]

J. BringerC. CarletH. ChabanneS. Guilley and H. Maghrebi, Orthogonal direct sum masking a smartcard friendly computation paradigm in a code with builtin protection against side-channel and fault attacks, WISTP, 8501 (1999), 40-56. 

[2]

C. CarletC. GüneriS. Mesnager and F. Özbudak, Construction of some codes suitable for both side channel and fault injection attacks, Arithmetic of Finite Fields, 11321 (2018), 95-107.  doi: 10.1007/978-3-030-05153-2_5.

[3]

C. CarletC. Li and S. Mesnager, Some (almost) optimally extendable linear codes, Des. Codes Cryptogr., 87 (2019), 2813-2834.  doi: 10.1007/s10623-019-00652-7.

[4]

P. Delsarte, On subfield subcodes of modifield Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.

[5]

C. Ding and C. Tang, Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory, 66 (2020), 5419-5428.  doi: 10.1109/TIT.2020.2990396.

[6]

Z. DuC. Li and S. Mesnager, Constructions of self-orthogonal codes from hulls of BCH codes and their parameters, IEEE Trans. Inf. Theory, 66 (2020), 6774-6785.  doi: 10.1109/TIT.2020.2991635.

[7]

K. Feng and F, Liu, Algebra and Communication, Higher Education Press, 2005.

[8]

Z. Heng and C. Ding, A construction of $q$-ary linear codes with irreducible cyclic codes, Des. Codes Cryptogr., 87 (2019), 1087-1108.  doi: 10.1007/s10623-018-0507-0.

[9]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[10]

W. C. Hufiman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.

[11]

D. Huang, Q. Yue and Y. Niu, MDS or NMDS LCD codes from twisted reed-solomon codes, Available from: https://seta2020.etu.ru/assets/files/program/paper/paper-43.pdf.

[12]

D. HuangQ. YueY. Niu and X. Li, MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes, Des. Codes Cryptogr., 89 (2021), 2195-2209.  doi: 10.1007/s10623-021-00910-7.

[13]

C. LiS. BaeJ. AhnS. Yang and Z. Yao, Complete weight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.  doi: 10.1007/s10623-015-0136-9.

[14]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[15] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. 
[16]

F. J. Macwilliams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977.

[17]

J. H. Van Lint, Introduction to Coding Theory, 3$^{rd}$ edition, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-58575-3.

[18]

Y. WuQ. YueX. Zhu and S. Yang, Weight enumerators of reducible cyclic codes and their dual codes, Discret. Math., 342 (2019), 671-682.  doi: 10.1016/j.disc.2018.10.035.

[19]

Y. WuQ. Yue and X. Shi, At most three-weight binary linear codes from generalized Moisio's exponential sums, Des. Codes Cryptogr., 87 (2019), 1927-1943.  doi: 10.1007/s10623-018-00595-5.

show all references

References:
[1]

J. BringerC. CarletH. ChabanneS. Guilley and H. Maghrebi, Orthogonal direct sum masking a smartcard friendly computation paradigm in a code with builtin protection against side-channel and fault attacks, WISTP, 8501 (1999), 40-56. 

[2]

C. CarletC. GüneriS. Mesnager and F. Özbudak, Construction of some codes suitable for both side channel and fault injection attacks, Arithmetic of Finite Fields, 11321 (2018), 95-107.  doi: 10.1007/978-3-030-05153-2_5.

[3]

C. CarletC. Li and S. Mesnager, Some (almost) optimally extendable linear codes, Des. Codes Cryptogr., 87 (2019), 2813-2834.  doi: 10.1007/s10623-019-00652-7.

[4]

P. Delsarte, On subfield subcodes of modifield Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.

[5]

C. Ding and C. Tang, Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory, 66 (2020), 5419-5428.  doi: 10.1109/TIT.2020.2990396.

[6]

Z. DuC. Li and S. Mesnager, Constructions of self-orthogonal codes from hulls of BCH codes and their parameters, IEEE Trans. Inf. Theory, 66 (2020), 6774-6785.  doi: 10.1109/TIT.2020.2991635.

[7]

K. Feng and F, Liu, Algebra and Communication, Higher Education Press, 2005.

[8]

Z. Heng and C. Ding, A construction of $q$-ary linear codes with irreducible cyclic codes, Des. Codes Cryptogr., 87 (2019), 1087-1108.  doi: 10.1007/s10623-018-0507-0.

[9]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[10]

W. C. Hufiman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.

[11]

D. Huang, Q. Yue and Y. Niu, MDS or NMDS LCD codes from twisted reed-solomon codes, Available from: https://seta2020.etu.ru/assets/files/program/paper/paper-43.pdf.

[12]

D. HuangQ. YueY. Niu and X. Li, MDS or NMDS self-dual codes from twisted generalized Reed-Solomon codes, Des. Codes Cryptogr., 89 (2021), 2195-2209.  doi: 10.1007/s10623-021-00910-7.

[13]

C. LiS. BaeJ. AhnS. Yang and Z. Yao, Complete weight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.  doi: 10.1007/s10623-015-0136-9.

[14]

C. LiQ. Yue and F. Li, Weight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28 (2014), 94-114.  doi: 10.1016/j.ffa.2014.01.009.

[15] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press, Cambridge, 1997. 
[16]

F. J. Macwilliams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977.

[17]

J. H. Van Lint, Introduction to Coding Theory, 3$^{rd}$ edition, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-642-58575-3.

[18]

Y. WuQ. YueX. Zhu and S. Yang, Weight enumerators of reducible cyclic codes and their dual codes, Discret. Math., 342 (2019), 671-682.  doi: 10.1016/j.disc.2018.10.035.

[19]

Y. WuQ. Yue and X. Shi, At most three-weight binary linear codes from generalized Moisio's exponential sums, Des. Codes Cryptogr., 87 (2019), 1927-1943.  doi: 10.1007/s10623-018-00595-5.

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