Abstract
In this paper, we clarify some aspects of LCD codes in the literature. We first prove that non-free LCD codes do not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD codes over a finite commutative Frobenius ring. We later show that a free constacyclic code over a finite chain ring is an LCD code if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible. We illustrate the minimum Lee distance of LCD codes over some finite commutative chain rings with examples. We found some new optimal cyclic codes over \({\mathbb {Z}}_4\) of different lengths which are LCD codes using computer algebra system MAGMA.
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Notes
\({\texttt {lcm}}(\pi (g_1), \pi (g_2))\): the least common multiple of \(\pi (g_1),\pi (g_2)\).
References
Aydin N., Asamov T.: http://www.asamov.com/Z4Codes/CODES/ShowCODESTablePage.aspx.
Boonniyoma K., Jitman S.: Complementary dual subfield linear codes over finite fields. ArXiv:1605.06827 [cs.IT] (2016).
Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. In: Pinto E.R., et al. (eds.) Coding Theory and Applications, vol. 3, pp. 97–105. CIM Series in Mathematical SciencesSpringer, Berlin (2014).
Carlet C., Güneri C., Özbudak F., Ozkaya B., Solé P.: On linear complementary pairs of codes. IEEE Trans. Inf. Theory 64(10), 6583–6589 (2018).
Carlet C., Mesnager S., Tang C.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2606–2618 (2018).
Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65, 39–49 (2018).
Carlet C., Mesnager S., Tang C., Qi Y., Pellikaan R.: Linear codes over \(F_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018). (Special Issue in honor of Solomon Golomb).
Dougherty S.T., Liu H.: Independence of vectors in codes over rings. Des. Codes Cryptogr. 51, 55–68 (2009).
Dougherty S.T., Kim J.L., Ozkaya B., Sok L., Solé P.: The combinatorics of LCD codes: linear programming bound and orthogonal matrices. To appear in Int. J. Inf. Coding Theory (IJICOT).
Dougherty S.T., Yildiz B., Karadeniz S.: Codes over \(R_k\), Gray maps and their binary images. Finite Fields Appl. 17, 205–219 (2011).
Fan Y., Ling S., Liu H.: Matrix product codes over finite commutative Frobenius rings. Des. Codes Cryptogr. 71, 201–227 (2014).
Fotue-Tabue A., Martínez-Moro E., Blackford T.: On polycyclic codes over a finite chain ring. Adv. Math. Commun. https://doi.org/10.3934/amc.2020028 (2019).
Gary M.G.: An approach to Hensel’s lemma. Ir. Math. Soc. Bull. 47, 15–21 (2001).
Güneri C., Özbudak F., Ozkaya B., Sacikara E., Sepasdar Z., Solé P.: Structure and performance of generalized quasi-cyclic codes. Finite Fields Appl. 47, 183–202 (2017).
Güneri C., Ozkaya B., Solé P.: Quasi-cyclic complementary dual codes. Finite Fields Appl. 42, 67–80 (2016).
Jin L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63(5), 2843–2847 (2017).
Kaplansky I.: Projective modules. Ann. Math. 68, 372–377 (1958).
Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017).
Li S., Li C., Ding C., Liu H.: Two families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017).
Liu X., Liu H.: LCD codes over finite chain rings. Finite Fields Appl. 34, 1–19 (2015).
López-Permouth S.R., Ozadam H., Özbudak F., Szabo S.: Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes. Finite Fields Appl. 19, 16–38 (2013).
Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992).
Massey J.L.: Reversible codes. Inf. Control 7(3), 369–380 (1964).
McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974).
Norton G.H., Salagean A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000).
Sendrier N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discret. Math. 285(1), 345–347 (2004).
Tzeng K., Hartmann C.: On the minimum distance of certain reversible cyclic codes. IEEE Trans. Inf. Theory 16(5), 644–646 (1970).
Wieb B., John C., Catherine P.: The Magma algebra system I. The user language, computational algebra and number theory (London, 1993). J. Symb. Comput. 24(3–4), 235–265 (1997).
Wood J.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121, 555–575 (1999).
Wu Y., Yue Q.: Factorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes. IEEE Trans. Inf. Theory 65(3), 1740–1751 (2019).
Yang X., Massey J.L.: The condition for a cyclic code to have a complementary dual. Discret. Math. 126, 391–393 (1994).
Acknowledgements
The first author of the paper would like to thank the Ministry of Human Resource and Development India for financial support to carry out this work. The third author is partially funded by the Spanish State Research Agency (AEI) under Grant PGC2018-096446-B-C21.
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Bhowmick, S., Fotue-Tabue, A., Martínez-Moro, E. et al. Do non-free LCD codes over finite commutative Frobenius rings exist?. Des. Codes Cryptogr. 88, 825–840 (2020). https://doi.org/10.1007/s10623-019-00713-x
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DOI: https://doi.org/10.1007/s10623-019-00713-x