# ON LCD CODES OVER FINITE CHAIN RINGS

• Accepted : 2019.10.10
• Published : 2020.01.31

#### Abstract

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. LCD cyclic codes have been known as reversible cyclic codes that had applications in data storage. Due to a newly discovered application in cryptography, interest in LCD codes has increased again. Although LCD codes over finite fields have been extensively studied so far, little work has been done on LCD codes over chain rings. In this paper, we are interested in structure of LCD codes over chain rings. We show that LCD codes over chain rings are free codes. We provide some necessary and sufficient conditions for an LCD code C over finite chain rings in terms of projections of linear codes. We also showed the existence of asymptotically good LCD codes over finite chain rings.

#### References

1. A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr. 6 (1995), no. 1, 21-35. https://doi.org/10.1007/BF01390768 https://doi.org/10.1007/BF01390768
2. C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun. 10 (2016), no. 1, 131-150. https://doi.org/10.3934/amc.2016.10.131 https://doi.org/10.3934/amc.2016.10.131
3. C. Carlet, S. Mesnager, C. M. Tang, Y. F. Qi, and R. Pellikaan, Linear codes over $F_q$ are equivalent to LCD codes for q > 3, IEEE Trans. Inform. Theory 64 (2018), no. 4, part 2, 3010-3017. https://doi.org/10.1109/TIT.2018.2789347 https://doi.org/10.1109/TIT.2018.2789347
4. S. T. Dougherty, J.-L. Kim, B. Ozkaya, L. Sok, and P. Sole, The combinatorics of LCD codes: linear programming bound and orthogonal matrices, Int. J. Inf. Coding Theory 4 (2017), no. 2-3, 116-128. https://doi.org/10.1504/IJICOT.2017.083827 https://doi.org/10.1504/IJICOT.2017.083827
5. M. Esmaeili and S. Yari, On complementary-dual quasi-cyclic codes, Finite Fields Appl. 15 (2009), no. 3, 375-386. https://doi.org/10.1016/j.ffa.2009.01.002 https://doi.org/10.1016/j.ffa.2009.01.002
6. Y. Fan, S. Ling, and H. Liu, Matrix product codes over finite commutative Frobenius rings, Des. Codes Cryptogr. 71 (2014), no. 2, 201-227. https://doi.org/10.1007/s10623-012-9726-y https://doi.org/10.1007/s10623-012-9726-y
7. C. Guneri, B. Ozkaya, and P. Sole, Quasi-cyclic complementary dual codes, Finite Fields Appl. 42 (2016), 67-80. https://doi.org/10.1016/j.ffa.2016.07.005 https://doi.org/10.1016/j.ffa.2016.07.005
8. M. Hazewinkel, Handbook of Algebra. Vol. 5, Handbook of Algebra, 5, Elsevier/North-Holland, Amsterdam, 2008.
9. T. Honold and I. Landjev, Linear codes over finite chain rings, Electron. J. Combin. 7 (2000), Research Paper 11, 22 pp. https://doi.org/10.37236/1500
10. L. Jin, Construction of MDS codes with complementary duals, IEEE Trans. Inform. Theory 63 (2017), no. 5, 2843-2847. https://doi.org/10.1109/TIT.2016.2644660 https://doi.org/10.1109/TIT.2016.2644660
11. C. Li, Hermitian LCD codes from cyclic codes, Des. Codes Cryptogr. 86 (2018), no. 10, 2261-2278. https://doi.org/10.1007/s10623-017-0447-0 https://doi.org/10.1007/s10623-017-0447-0
12. X. Liu and H. Liu, LCD codes over finite chain rings, Finite Fields Appl. 34 (2015), 1-19. https://doi.org/10.1016/j.ffa.2015.01.004 https://doi.org/10.1016/j.ffa.2015.01.004
13. J. L. Massey, Reversible codes, Information and Control 7 (1964), 369-380. https://doi.org/10.1016/S0019-9958(64)90438-3
14. J. L. Massey, Linear codes with complementary duals, Discrete Math. 106/107 (1992), 337-342. https://doi.org/10.1016/0012-365X(92)90563-U https://doi.org/10.1016/0012-365X(92)90563-U
15. B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974.
16. S. Mesnager, C. Tang, and Y. Qi, Complementary dual algebraic geometry codes, IEEE Trans. Inform. Theory 64 (2018), no. 4, part 1, 2390-2397. https://doi.org/10.1109/TIT.2017.2766075 https://doi.org/10.1109/TIT.2017.2766075
17. G. H. Norton and A. Salagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory 46 (2000), no. 3, 1060-1067. https://doi.org/10.1109/18.841186 https://doi.org/10.1109/18.841186
18. G. H. Norton, On the structure of linear and cyclic codes over a finite chain ring, Appl. Algebra Engrg. Comm. Comput. 10 (2000), no. 6, 489-506. https://doi.org/10.1007/PL00012382 https://doi.org/10.1007/PL00012382
19. N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math. 285 (2004), no. 1-3, 345-347. https://doi.org/10.1016/j.disc.2004.05.005 https://doi.org/10.1016/j.disc.2004.05.005
20. L. Sok, M. Shi, and P. Sole, Constructions of optimal LCD codes over large finite fields, Finite Fields Appl. 50 (2018), 138-153. https://doi.org/10.1016/j.ffa.2017.11.007 https://doi.org/10.1016/j.ffa.2017.11.007
21. J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555-575. https://doi.org/10.1353/ajm.1999.0024 https://doi.org/10.1353/ajm.1999.0024
22. X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math. 126 (1994), no. 1-3, 391-393. https://doi.org/10.1016/0012-365X(94)90283-6 https://doi.org/10.1016/0012-365X(94)90283-6
23. H. Zhu and M. Shi, On linear complementary dual four circulant codes, Bull. Aust. Math. Soc. 98 (2018), no. 1, 159-166. https://doi.org/10.1017/S0004972718000175 https://doi.org/10.1017/S0004972718000175