具有l维Hermitian正交包的MDS码的构造
Construction of MDS Code with l-Dimensional Hermitian Hull
摘要: 达到 Singleton 界的码称为极大距离可分码(简称为 MDS 码)，其纠错能力最强，在纠错码中有着非常广泛的应用。本文研究了MDS码的Hermitian正交包，利用广义 Reed-Solomon 码构造 了具有l(l ≥ 1)维Hermitian正交包的MDS码。
Abstract: The codes achieving the Singleton bound are called maximum distance separable (for short MDS) codes, which have the strongest error correction ability and are widely applied in error-correcting code. In this paper, we study the Hermitian hulls of MDS codes.  We use the generalized Reed-Solomon code to construct MDS codes with l-dimensional (l ≥ 1) Hermitian hull.
文章引用：韩雨慧, 邱宇廷, 卢啸华. 具有l维Hermitian正交包的MDS码的构造[J]. 理论数学, 2020, 10(11): 1015-1024. https://doi.org/10.12677/PM.2020.1011120

参考文献

[1] Leon, J. (1982) Computing Automorphism Groups of Error-Correcting Codes. IEEE Transac- tions on Information Theory, 28, 496-511.
https://doi.org/10.1109/TIT.1982.1056498
[2] Sendrier, N. and Skersys, G. (2001) On the Computation of the Automorphism Group of a Linear Code. Proceedings of the 2001 IEEE International Symposium on Information Theory, Washington DC, 29 June 2001, 13.
https://doi.org/10.1109/ISIT.2001.935876
[3] Brun, T., Devetak, I. and Hsieh, M.H. (2006) Correcting Quantum Errors with Entanglement. Science, 314, 436-439.
https://doi.org/10.1126/science.1131563
[4] Sendrier, N. (1997) On the Dimension of the Hull. SIAM Journal on Discrete Mathematics, 10, 282-293.
https://doi.org/10.1137/S0895480195294027
[5] Sangwisut, E., Jitman, S., Ling, S. and Udomkavanich, P. (2015) Hulls of Cyclic and Negacyclic Codes over Finite Fields. Finite Fields and Their Applications, 33, 232-257.
https://doi.org/10.1016/j.ffa.2014.12.008
[6] Skersys, G. (2003) The Average Dimension of the Hull of Cyclic Codes. Discrete Applied Mathematics, 128, 275-292.
https://doi.org/10.1016/S0166-218X(02)00451-1
[7] Jitman, S. and Sangwisut, E. (2018) The Average Dimension of the Hermitian Hull of Consta- cyclic Codes over Finite Fields of Square Order. Advances in Mathematics of Communications, 12, 451-463.
https://doi.org/10.3934/amc.2018027
[8] Jin, L. (2016) Construction of MDS Codes with Complementary Duals. IEEE Transactions on Information Theory, 63, 2843-2847.
https://doi.org/10.1109/TIT.2016.2644660
[9] Beelen, P. and Jin, L. (2018) Explicit MDS Codes with Complementary Duals. IEEE Trans- actions on Information Theory, 64, 7188-7193.
https://doi.org/10.1109/TIT.2018.2816934
[10] Luo, G. and Cao, X. (2018) MDS Codes with Arbitrary Dimensional Hull and Their Applica- tions. arXiv:1807.03166
[11] Chen, B. and Liu, H. (2017) New Constructions of MDS Codes with Complementary Duals. IEEE Transactions on Information Theory, 64, 5776-5782.
https://doi.org/10.1109/TIT.2017.2748955
[12] Carlet, C., Mesnager, S., Tang, C. and Qi, Y. (2018) Euclidean and Hermitian LCD MDS Codes. Designs, Codes and Cryptography, 86, 2605-2618.
https://doi.org/10.1007/s10623-018-0463-8
[13] Fang, W., Fu, F.W., Li, L. and Zhu, S. (2020) Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs. IEEE Transactions on Information Theory, 66, 3527- 3537.
https://doi.org/10.1109/TIT.2019.2950245
[14] Li, C. (2018) Hermitian LCD Codes from Cyclic Codes. Designs, Codes and Cryptography, 86, 2261-2278.
https://doi.org/10.1007/s10623-017-0447-0
[15] Ding, Y. and Lu, X.H. (2020) Galois Hulls of Cyclic Codes over Finite Fields. IEICE Trans- actions on Communication, 103, 370-375.
https://doi.org/10.1587/transfun.2019EAL2087
[16] MacWilliams, F.J. and Sloane, N.J.A. (1977) The Theory of Error-Correcting Codes. Elsevier, Amsterdam.

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