Abstract
Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we outline a construction for LCD codes over finite fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Under certain conditions, the method yields formally self-dual codes. Further, we propose a decoding algorithm that can be feasible for the LCD codes obtained using one of the given methods.
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References
Alahmadi, A., Altassan, A., AlKenani, A., Çalkavur, S., Shoaib, H., Solé, P.: A multisecret-sharing scheme based on LCD codes. Mathematics 8(2), 272 (2020)
Alahmadi, A., Deza, M., Sikirić, M.D., Solé, P.: The joint weight enumerator of an LCD code and its dual. Discrete Appl. Math. 257, 12–18 (2019)
Araya, M., Harada, M.: On the classification of linear complementary dual codes. Discrete Math. 342, 270–278 (2019)
Assmus, E.F., Key, J.D.: Designs and their Codes. Cambridge University Press, Cambridge (1992)
Bosma, W., Cannon, J.: Handbook of Magma Functions, Department of Mathematics, University of Sydney, (1994). http://magma.maths.usyd.edu.au/magma
Carlet, C., Guilley, S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10, 131–150 (2016)
Carlet, C., Mesnager, S., Tang, C., Qi, Y.: Linear codes over \(F_q\) which are equivalent to LCD codes. IEEE Trans. Inform. Theory 64, 3010–3017 (2018)
Crnković, D., Egan, R., Rodrigues, B.G., Švob, A.: LCD codes from weighing matrices. Appl. Algebra Eng. Commun. Comput. (2019). https://doi.org/10.1007/s00200-019-00409-8
Crnković, D., Mikulić, V.: Self-orthogonal doubly-even codes from Hadamard matrices of order 48. Adv. Appl. Discrete Math. 1, 159–170 (2008)
Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Reports Suppl. 10, (1973)
Dougherty, S.T., Kim, J.-L., Solé, P.: Double circulant codes from two-class association schemes. Adv. Math. Commun. 1, 45–64 (2007)
Gaborit, P.: Quadratic double circulant codes over fields. J. Combin. Theory Ser. A 97, 85–107 (2002)
Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes, http://www.codetables.de. Accessed (20 January 2020)
Harada, M., Saito, K.: Binary linear complementary dual codes. Cryptogr. Commun. 11, 677–696 (2019)
Higman, D.G.: Coherent configuration. Geom. Dedicata 4, 1–32 (1975)
Horiguchi, N., Nakasora, H., Wakabayashi, T.: On the strongly regular graphs obtained from quasi-symmetric 2-(31,7,7) designs. Bull. Yamagata Univ. Natur. Sci. 16, 1–6 (2005)
Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Kennedy, G.T., Pless, V.: On designs and formally self-dual codes. Des. Codes Cryptogr. 4, 43–55 (1994)
Key, J.D., Rodrigues, B.G.: Special LCD codes from Peisert and Generalized Peisert Graphs. Graphs Combin. 35, 633–652 (2019)
Koukouvinos, C.: http://www.math.ntua.gr/~ckoukouv/. Accessed (20 January 2020)
Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106(107), 337–342 (1992)
Qian, L., Shi, M., Solé, P.: On self-dual and LCD quasi-twisted codes of index two over a special chain ring. Cryptogr. Commun. 11, 717–734 (2019)
Reid, K.B., Brown, E.: Doubly regular tournaments are equivalent to skew Hadamard matrices. J. Combin. Theory Ser. A 12, 332–338 (1972)
Sendrier, N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discrete Math. 304, 345–347 (2004)
Sok, L., Shi, M., Solé, P.: Constructions of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018)
Shi, M., Zhu, H., Qian, L., Sok, L., Solé, P.: On self-dual and LCD double circulant and double negacirculant codes over Fq+uFq. Cryptogr. Commun. 12, 53–70 (2020)
Shi, M., Huang, D., Sok, L., Solé, P.: Double circulant LCD codes over \(Z_4\). Finite Fields Appl. 58, 133–144 (2019)
Stoichev, S.D.: The nonisomorphism of the strongly regular graphs derived from the quasi-symmetric 2-(31,7,7) designs. C. R. Acad. Bulgare Sci. 40, 33–35 (1987)
Tonchev, V.D.: Quasi-symmetric 2-(31,7,7) designs and a revision of Hamada’s conjecture. J. Comb. Theory Ser. A 42, 104–110 (1986)
Wu, R., Shi, M.: A modified Gilbert-Varshamov bound for self-dual quasi-twisted codes of index four. Finite Fields Appl. 62, 101627 (2020)
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The authors thank the anonymous referees for helpful suggestions that improved the paper.
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This work has been fully supported by Croatian Science Foundation under the project 6732.
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Crnković, D., Grbac, A. & Švob, A. Formally self-dual LCD codes from two-class association schemes. AAECC 34, 183–200 (2023). https://doi.org/10.1007/s00200-021-00497-5
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DOI: https://doi.org/10.1007/s00200-021-00497-5