# American Institute of Mathematical Sciences

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:
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##### References:
 [1] J. Bringer, C. Carlet, H. Chabanne, S. 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. Carlet, C. Güneri, S. 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. Carlet, C. 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. Du, C. 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. Heng, C. 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. Huang, Q. Yue, Y. 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. Li, S. Bae, J. Ahn, S. 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. Li, Q. 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. Wu, Q. Yue, X. 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. Wu, Q. 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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