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Construction of LCD codes from tridiagonal Toeplitz matrices | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 09 تیر 1405 اصل مقاله (461.7 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.30942.2674 | ||
| نویسندگان | ||
| Wajid M. Shaikh1، 2؛ Rupali S. Jain1؛ B. Surendranath Reddy* 1؛ Bhagyashri S. Patil1، 3 | ||
| 1School of Mathematical Sciences, SRTMU Nanded, India | ||
| 2P.E.S. College of Engineering, Chhatrapati Sambhajinagar, India | ||
| 3MGM’s College of Engineering, Nanded, India | ||
| چکیده | ||
| A Toeplitz matrix $T$ is characterized by having constant entries along diagonals parallel to the main diagonal. Double Toeplitz (DT) codes are linear codes whose generator matrix takes the form $(I, T)$, where $T$ is a Toeplitz matrix. In 2021, Shi et al. established the necessary and sufficient condition for a DT code to be an LCD, assuming that $T$ is symmetric. In 2024, Cheng obtained the necessary and sufficient condition for a DT code to be an LCD when $T$ is skew-symmetric. In this paper, we consider Toeplitz tridiagonal matrices that are neither symmetric nor skew-symmetric. We derive the necessary and sufficient condition under which a DT code is an LCD code, using the factorization of Dickson polynomials over finite fields. Furthermore, by applying concatenation techniques, we construct a family of LCD codes with arbitrary minimum distance. | ||
| کلیدواژهها | ||
| LCD code؛ Toeplitz matrix؛ Dickson polynomial | ||
| مراجع | ||
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[1] A. Agrawal, G.K. Verma, and R.K. Sharma, Galois LCD codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, Bull. Aust. Math. Soc. 107 (2023), no. 2, 330–341. https://doi.org/10.1017/S0004972722001344
[2] M. Bhargava and M.E. Zieve, Factoring Dickson polynomials over finite fields, Finite Fields Appl. 5 (1999), no. 2, 103–111. https://doi.org/10.1006/ffta.1998.0221
[3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. https://doi.org/10.1006/jsco.1996.0125
[4] C. Carlet and S. Guilley, Complementary dual codes for counter-measures to sidechannel attacks, Coding Theory and Applications (R. Pinto, P. Rocha Malonek, and P. Vettori, eds.), CIM Series in Mathematical Sciences, vol. 3, Springer, Cham, 2014, pp. 97–105. https://doi.org/10.1007/978-3-319-17296-5_9 [5] C. Carlet, C. Güneri, F. Özbudak, and P. Solé, A new concatenated type construction for LCD codes and isometry codes, Discrete Math. 341 (2018), no. 3, 830–835. https://doi.org/10.1016/j.disc.2017.12.004
[6] C. Carlet, S. Mesnager, C. Tang, and Y. Qi, Euclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr. 86 (2018), 2605–2618. https://doi.org/10.1007/s10623-018-0463-8
[7] C. Carlet, S. Mesnager, C. Tang, and Y. Qi, On $\sigma$-LCD codes, IEEE Transactions on Information Theory 65 (2019), no. 3, 1694–1704. https://doi.org/10.1109/TIT.2018.2873130 [8] C. Carlet, S. Mesnager, C. Tang, Y. Qi, and R. Pellikaan, Linear codes over $\mathbb{F}_q$ are equivalent to LCD codes for $q > 3$, IEEE Transactions on Information Theory 64 (2018), no. 4, 3010–3017. https://doi.org/10.1109/TIT.2018.2789347
[9] K. Cheng, On LCD codes from skew symmetric Toeplitz matrices, Finite Fields Appl. 95 (2024), 102380. https://doi.org/10.1016/j.ffa.2024.102380 [10] D.T. Huang, M.J. Shi, and P. Solé, Double circulant self-dual and LCD codes over $\mathbb{Z}_{p^2}$ , Internat. J. Found. Comput. Sci. 30 (2019), no. 3, 407–416. https://doi.org/10.1142/S0129054119500114
[11] S. Li, M. Shi, and J. Wang, An improved method for constructing formally selfdual codes with small hulls, Des. Codes Cryptogr. 91 (2023), no. 7, 2563–2583. https://doi.org/10.1007/s10623-023-01210-y
[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
[13] Z. Liu and J. Wang, Linear complementary dual codes over rings, Des. Codes Cryptogr. 87 (2019), 3077–3086. https://doi.org/10.1007/s10623-019-00664-3 [14] Z. Liu and X.W. Wu, Notes on LCD codes over frobenius rings, IEEE Communications Letters 25 (2021), no. 2, 361–364. https://doi.org/10.1109/LCOMM.2020.3029073
[15] J.L. Massey, Linear codes with complementary duals, Discrete Math. 106 (1992), 337–342. https://doi.org/10.1016/0012-365X(92)90563-U
[16] N. Sendrier, On the dimension of the hull, SIAM J. Discrete Math. 10 (1997), no. 2, 282–293. https://doi.org/10.1137/S0895480195294027 [17] N. Sendrier, Finding the permutation between equivalent linear codes: The support splitting algorithm, IEEE Transactions on Information Theory 46 (2000), no. 4, 1193–1203. https://doi.org/10.1109/18.850662
[18] M. Shi, D. Huang, L. Sok, and P. Solé, Double circulant LCD codes over $\mathbb{Z}_4$, Finite Fields Appl. 58 (2019), 133–144. https://doi.org/10.1016/j.ffa.2019.04.001 [19] M. Shi, D. Huang, L. Sok, and P. SoléDouble circulant self-dual and LCD codes over Galois rings, Adv. Math. Commun. 13 (2019), no. 1, 171–183. https://doi.org/10.3934/amc.2019011
[20] M. Shi, F. Özbudak, L. Xu, and P. Solé, LCD codes from tridiagonal Toeplitz matrices, Finite Fields Appl. 75 (2021), 101892. https://doi.org/10.1016/j.ffa.2021.101892
21] M. Shi, L. Qian, and P. Solé, On the self-dual negacirculant codes of index two and four, Des. Codes Cryptogr. 86 (2018), no. 11, 2485–2494. https://doi.org/10.1007/s10623-017-0455-0
[22] M. Shi, L. Xu, and P. Solé, On isodual double Toeplitz codes, J. Sys. Sci. Complex. 37 (2024), no. 5, 2196–2206. https://doi.org/10.1007/s11424-024-2397-8 [23] M. Shi, H. Zhu, L. Qian, L. Sok, and P. Solé, On self-dual and LCD double circulant and double negacirculant codes over $\mathbb{F}_q + u\mathbb{F}_q$, Cryptogr. Commun. 12 (2020), no. 1, 53–70. https://doi.org/10.1007/s12095-019-00363-9$ | ||
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