Abstract
In this paper, we show that LCD codes are not equivalent to non-LCD codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD(n,2) over \(\mathbb {F}_{3}\) and \(\mathbb {F}_{4}\), where LD(n,2) := max{d∣thereexsitsan [n,2, d] LCD\( code~ over~ \mathbb {F}_{q}\}\). We study the bound of LCD codes over \(\mathbb {F}_{q}\) and generalize a conjecture proposed by Galvez et al. about the minimum distance of binary LCD codes.
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References
Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Comm. 10(1), 131–150 (2016)
Carlet, C., Güneri, C., Özbudak, F., Solé, P.: A new concatenated type construction for LCD codes and isometry codes. Discrete Math. 341(3), 830–835 (2018)
Carlet, C., Mesnager, S., Tang, C.M., Qi, Y.F.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39–49 (2019)
Carlet, C., Mesnager, S., Tang, C.M., Qi, Y.F.: On σ-LCD codes. IEEE Trans. Inf. Theory 65(3), 1694–1704 (2019)
Carlet, C., Mesnager, S., Tang, C.M., Qi, Y.F.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2605–2618 (2018)
Carlet, C., Mesnager, S., Tang, C.M., Qi, Y.F., Pellikaan, R.: Linear codes over \(\mathbb {F}_{q}\) which are equivalent to LCD codes for q > 3. IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018)
Dougherty, S.T., Kim, J.L., Ozkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: Linear programming bound and orthogonal matrices. Int. J. Inf. Coding Theory 4(2-3), 116–128 (2017)
Ding, C.S., Li, C.J., Li, S.X.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017)
Dinh, H.Q., Nguyen, B.T., Sriboonchitta, S.: Constacyclic codes over finite commutative semi-simple rings. Finite Fields Their Appl. 45, 1–18 (2017)
Galvez, L., Kim, J.L., Lee, N., Roe, Y.G., Won, B.S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10(4), 719–728 (2018)
Geri, C., Özkaya, B., Solé, P.: Quasi-cyclic complementary dual codes. Finite Fields Their Appl. 42, 67–80 (2016)
Huffman, W.C., Pless, V.: Fundamentals of Error-correcting Codes. Cambridge University Press, Cambridge (2010)
Harada, M., Saito, K.: Binary linear complementary dual codes. Cryptogr. Commun. 11(4), 677–696 (2019)
Jin, L.F.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843–2847 (2017)
Jin, L.F., Xing, C.P.: Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound. IEEE Trans. Inf. Theory 64(9), 6277–6282 (2018)
Körŏglu, M.E., Sari, M.: On MDS negacyclic LCD codes. Filomat 31(1), 1–12 (2019)
Liu, X.S., Fan, Y., Liu, H.L.: Galois LCD codes over finite fields. Finite Fields Their Appl. 49, 227–242 (2018)
Liu, X.S., Liu, H.: LCD codes over finite chain rings. Finite Fields Their Appl. 34, 1–19 (2015)
Liu, X.S., Liu, H.: Matrix-Product complementary dual codes, arXiv:1604.03774 (2016)
Li, S.X., Li, C.J., Ding, C.S., Liu, H.: Two families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017)
Lina, E.R.J., Nocon, E.G.: On the construction of some LCD codes over finite fields. Manila J. Sci. 9, 67–82 (2016)
Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106(/107), 337–342 (1992)
Mesnager, S., Tang, C.M., Qi, Y.F.: Complementary dual algebraic geometry codes. IEEE Trans. Inf. Theory 64(4), 2390–2397 (2018)
Pang, B.B., Zhu, S.X., Sun, Z.H.: On LCD negacyclic codes over finite fields. J. Syst. Sci. Complex. 31(4), 1065–1077 (2018)
Rao, Y., Li, R.H., Lv, L.D., Chen, G., Zuo, F.: On binary LCD cyclic codes. Procedia Comput. Sci. 107, 778–783 (2017)
Sendrier, N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discrete Math. 285, 345–347 (2004)
Sok, L., Shi, M.J., Solé, P.: Construction of optimial LCD codes over larger finite fields. Finite Fields and Their Appl. 50, 138–153 (2018)
Yang, X., Massey, J.L.: The condition for a cyclic code to have a complementary dual. Discrete Math. 126(1–3), 391–393 (1994)
Acknowledgments
This research was supported by the National Natural Science Foundation of China Grant Nos 61772168 and 11501156 and the Fundamental Research Funds for the Central Universities of China Grant No. PA2019GDZC0097. The authors would like to thank the Editor-in-Chief Claude Carlet and anonymous referees for the useful comments.
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Pang, B., Zhu, S. & Kai, X. Some new bounds on LCD codes over finite fields. Cryptogr. Commun. 12, 743–755 (2020). https://doi.org/10.1007/s12095-019-00417-y
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DOI: https://doi.org/10.1007/s12095-019-00417-y