Lifted MDS Codes over Finite Fields

Year 2022, Volume: 7 Issue: 1, 45 - 49, 30.04.2022

Elif Segah Oztas

https://doi.org/10.30931/jetas.1032180

Abstract

MDS codes are elegant constructions in coding theory and have mode important applications in network coding, distributed data storage, communication systems etc. In this study, a method is given which MDS codes are lifted to a higher finite field. The presented method satisfies the protection of the distance and creating the MDS code over the $F_q$ by using MDS code over $F_p.$

Keywords

Lifted MDS codes , distance preserving , Linear code

References

• [1] MacWilliams, F. J., Sloane, N. J. A., "The theory of error-correcting codes", Amsterdam: North-Holland (1977).
• [2] Hurley, T., "MDS codes over finitefields", arXiv:1903.05265 [cs.IT].
• [3] Gupta, K. C., Pandey, S. K., Ray, I. G., Samanta, S. "Crypto- graphically significant mds matrices over finite fields: A brief survey and some generalized results", Advances in Mathematics of Communications 13 (4) (2019) : 779-843.
• [4] Daemen, J., Rijmen, V., "The design of Rijndael: AES - The Advanced Encryption Standard, Springer-Verlag (2002).
• [5] Hu, Y., Zhang, X., Lee, P. P. C., Zhou, P., "Generalized optimal storage scaling via network coding", IEEE International Symposium on Information Theory - Proceedings, 2018-June, art. no. 8437684 (2018) : 956-960.
• [6] Guang, X., Yeung, R.W., "Linear network error correction coding revisited", IEEE International Symposium on Information Theory - Proceedings, 2020-June, art. no. 9174493 (2020) : 1635-1640.
• [7] Wu, X., Li, Q., Leung, V. C. M., Ching, P. C., "Joint Fronthaul multicast and cooperative beam- forming for cache-enabled cloud-based small cell networks: An MDS codes-aided approach", IEEE Transactions on Wireless Communications 18(10) art. no. 8786923 (2019) : 4970-4982.
• [8] Ko, D., Hong, B., Choi, W. "Probabilistic caching based on maximum distance separable code in a user-centric clustered cache-aided wireless network", IEEE Transactions on Wireless Communications 18(3) art. no. 8638790 (2019) : 1792-1804.
• [9] Gu, S., Li, J., Wang, Y., Wang, N., Zhang, Q. "DR-MDS: An energy-efficient coding scheme in D2D distributed storage network for the internet of things", IEEE Access 7 art. no. 8648336 (2019) : 24179-24191.
• [10] Pedersen, J., Graell Amat, A.I., Andriyanova, I., Brannstrom, F. "Optimizing MDS coded caching in wireless networks with device-to-device communication", IEEE Transactions on Wireless Communications 18(1) art. no. 8551275 (2019) : 286-295.
• [11] Mousavi, S., Zhou, T., Tian, C. "Delayed parity generation in MDS storage codes", IEEE International Symposium on Information Theory - Proceedings 8437700 (2018) : 1889-1893.
• [12] Heidarpour, A. R., Ardakani, M., Tellambura, C., "Network coded cooperation based on relay selection with imperfect CSI", IEEE Vehicular Technology Conference 2017-September, (2018) : 1-5.
• [13] Tamo, I., Wang, Z., Bruck, J. "Zigzag codes: MDS array codes with optimal rebuilding", IEEE Transactions on Information Theory 59(3) art. no. 6352912 (2013) : 1597-1616.
• [14] Guang, X., Fu, F.-W., Zhang, Z. "Construction of network error correction codes in packet networks", IEEE Transactions on Information Theory 59(2) art. no. 6320693, (2013) : 1030-1047.
• [15] Shah, N. B., Rashmi, K. V., Kumar, P. V., Ramchandran, K. "Interference alignment in regenerating codes for distributed storage: Necessity and code constructions", IEEE Transactions on Information Theory 5(4) art. no. 6096412 (2012) : 2134-2158.
• [16] Hu, Y., Xu, Y., Wang, X., Zhan, C., Li, P. "Cooperative recovery of distributed storage systems from multiple losses with network coding", IEEE Journal on Selected Areas in Communica tions 28 (2) art. no. 5402494 (2010) : 268-276.
• [17] Silva, D., Kschischang, F. R. "Security for wiretap networks via rank-metric codes", IEEE International Symposium on Information Theory - Proceedings art. no. 4594971 (2008) : 176-180.
• [18] Fragouli, C., Soljanin, E. "Information flow decomposition for network coding", IEEE Trans- actions on Information Theory 52(3) (2006) : 829-848.
• [19] Gupta, K. C., Ray, I. G., "Cryptographically significant MDS matrices based on circulant and circulant-like matrices for lightweight applications", Cryptography and Communications 7 (2015) : 257-287.
• [20] Liu, M., Sim, S. M., "Lightweight MDS generalized circulant matrices", International Conference on Fast Software Encryption, Lecture Notes in Computer Science 9783 Springer, Berlin, Heidelberg (2016) : 101-120.
• [21] Sarkar, S., Syed, H., "Lightweight diffusion layer: Importance of Toeplitz matrices", IACR Trans. Symmetric Cryptol. (2016) : 95-113.
• [22] Sarkar, S., Syed, H., "Analysis of toeplitz MDS matrices", ACISP 2017, LNCS 10343 (2017) : 3-18.