Abstract
In this paper, a necessary condition which is sufficient as well for a pair of constacyclic 2-D codes over a finite commutative ring R to be an LCP of codes has been obtained. Also, a characterization of non-trivial LCP of constacyclic 2-D codes over R has been given and total number of such codes has been determined. The above results on constacyclic 2-D codes have been extended to constacyclic 3-D codes over R. The obtained results readily extend to constacyclic n-D codes, \(n \ge 3\), over finite commutative rings. Furthermore, some results on existence of non-trivial LCP of constacyclic 2-D codes over a finite chain ring have been obtained in terms of its residue field.
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References
Massey, J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992)
Bringer, J., Carlet, C., Chabanne, H., Guilley, S., Maghrebi, H.: Orthogonal direct sum masking: a smartcard friendly computation paradigm in a code with builtin protection against side-channel and fault attacks. WISTP, Heraklion, Crete, LNCS, vol. 8501, pp. 40–56. Springer, Berlin, Heidelberg (2014)
Ngo, X.T., Bhasin, S., Danger, J.L., Guilley, S., Najm, Z.: Linear complementary dual code improvement to strengthen encoded circuit against hardware Trojan horses. In: IEEE International Symposium on Hardware Oriented Security and Trust (HOST), Washington, pp. 82–87 (2015)
Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. J. Adv. Math. Commun. 10, 131–150 (2016)
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, 3010–3017 (2018)
Liu, X., Liu, H.: LCD codes over finite chain rings. Finite Fields Appl. 34, 1–19 (2015)
Carlet, C., Mesnager, S., Tang, C., Qi, Y.: On \(\sigma\) -LCD codes. IEEE Trans. Inf. Theory 65, 1694–1704 (2019)
Liu, X., Liu, H.: \(\sigma\)-LCD codes over finite chain rings. Des. Codes Cryptogr. 88, 727–746 (2020)
Dinh, H.Q., Yadav, B.P., Bag, T., et al.: Self-dual and LCD double circulant and double negacirculant codes over a family of finite rings \(F_{q}[v_1, v_2,\ldots , v_t]\). Cryptogr. Commun. 15, 529–551 (2022)
Carlet, C., Güneri, C., Özbudak, F., Özkaya, B., Solé, P.: On linear complementary pair of codes. IEEE Trans. Inf. Theory 64(10), 6583–6589 (2017)
Güneri, C., Özkaya, B., Sayıcı, S.: On linear complementary pair of nD cyclic codes. IEEE Commun. Lett. 22, 2404–2406 (2018)
Borello, M., de Cruz, J., Willems, W.: A note on linear complementary pairs of group codes. Discret. Math. 343, 111905 (2020)
Güneri, C., Martínez-Moro, E., Sayıcı, S.: Linear complementary pair of group codes over finite chain rings. Des. Codes Cryptogr. 88, 2397–2405 (2020)
Liu, X., Liu, H.: LCP of group codes over finite Frobenius rings. Des. Codes Cryptogr. 91, 695–708 (2022)
Hu, P., Liu, X.: Linear complementary pairs of codes over rings. Des. Codes Cryptogr. 89, 2495–2509 (2021)
Thakral, R., Dutt, S., Sehmi, R.: LCP of constacyclic codes over finite chain rings. J. Appl. Math. Comput. 69, 1989–2001 (2022)
Imai, H.: A theory of two-dimensional cyclic codes. Inf. Control 34(1), 1–21 (1977)
Güneri, C., Özbudak, F.: A relation between quasi-cyclic codes and 2-D cyclic codes. Finite Fields Appl. 18, 123–132 (2012)
Li, X., Hongyanb: \(2-D\) skew-cyclic codes over \(F_{q}[x, y,\rho ,\theta ]\). Finite Fields Appl. 25, 49–63 (2014)
Sepasdar, Z., Khashyarmanesh, K.: Characterizations of some two-dimensional cyclic codes correspond to the ideals of \(F[x, y]/ \langle x^{s} -1, y^{2^k}-1\rangle\). Finite Fields Appl. 41, 97–112 (2016)
Rajabi, Z., Khashyarmanesh, K.: Repeated-root two-dimensional constacyclic codes of length \(2p^s2^k\). Finite Fields Appl. 50, 122–137 (2018)
Hajiaghajanpour, N.: Khashyarmanesh, K: Two dimensional double cyclic codes over finite fields. In: AAECC (2023)
Bhardwaj, S., Raka, M.: Two dimensional constacyclic codes of arbitrary length over finite fields. Indian J. Pure Appl. Math. 53, 49–61 (2022)
Bhardwaj, S., Raka, M.: Multi-dimensional constacyclic codes of arbitrary length over finite fields. arXiv:2201.01031v1 [cs.IT] (2022)
Disha, D.S.: An algorithm to find the generators of multidimensional cyclic codes over a finite chain ring. arxiv:2303.07881v1 [cs.IT] (2023)
McDonald, B.R.: Finite rings with identity. Marcel Dekker Press, New York (1974)
Dinh, H.Q., Nguyen, H.D.T., Sriboonchitta, S., Vo, T.M.: Repeated root constacyclic codes of prime power lengths over finite chain rings. Finite Fields Appl. 43, 22–41 (2017)
Sharma, A., Sidana, T.: On the structure and distances of repeated-root constacyclic codes of prime power lengths over finite commutative chain rings. IEEE Trans. Inf. Theory 65, 2 (2019)
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Thakral, R., Dutt, S. & Sehmi, R. Linear complementary pairs of constacyclic n-D codes over a finite commutative ring. AAECC (2024). https://doi.org/10.1007/s00200-023-00640-4
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DOI: https://doi.org/10.1007/s00200-023-00640-4