Abstract
As far as we know, there are many different types of coding bounds in classical coding theory, which measure the efficiency of a code. Similarly, some important and useful coding bounds in classical coding theory are generalized to linear network error correction coding, including the Hamming bound, the Gilbert-Varshamov bound and the Singleton bound. While the former two bounds are both interesting theoretical results, the Singleton bound plays a very important role in the theory of network error correction. In this chapter, we first discuss the Hamming bound and the Singleton bound in detail. In particular, the LNEC maximum distance separable (MDS) codes, defined as the codes meeting the Singleton bound with equality, are studied in detail due to their optimality. Furthermore, we present several constructive algorithms of LNEC codes, particularly, for LNEC MDS codes, and analyze their performance.
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References
N. Cai and R. W. Yeung, “Network coding and error correction,” 2002 IEEE Information Theory Workshop, Bangalore, India, Oct. 20–25, 2002.
N. Cai and R. W. Yeung, “Network error correction, part II: Lower bounds,” Comm. Info. and Syst., 6: 37–54, 2006.
X. Guang, F.-W. Fu, and Z. Zhang, “Construction of Network Error Correction Codes in Packet Networks”, NetCod 2011, Beijing, China, July 25–27, 2011.
X. Guang, F.-W. Fu, and Z. Zhang, “Construction of Network Error Correction Codes in Packet Networks,” IEEE Trans. Info. Theory, IT-59: 1030–1047, 2013.
S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain, and L. M. G. M. Tolhuizen, “Polynomial time algorithms for multicast network code construction,” IEEE Trans. Info. Theory, IT-51: 1973–1982, 2005.
R. Matsumoto, “Construction algorithm for network error-correcting codes attaining the singleton bound,” IEICE Trans. Fundamentals, vol. E90-A, no. 9, pp. 1729–1735, Nov. 2007.
S. Yang, “Coherent network error correction,” Ph.D. dissertation, The Chinese University of Hong Kong, 2008.
S. Yang, R. W. Yeung, and C. K. Ngai, “Refined Coding Bounds and Code Constructions for Coherent Network Error Correction,” IEEE Trans. Info. Theory, IT-57: 1409–1424, 2011.
R. W. Yeung, Information Theory and Network Coding. New York: Springer, 2008.
R. W. Yeung and N. Cai, “Network error correction, part I: Basic concepts and upper bounds,” Communications in Information and Systems, 6: 19–36, 2006.
R. W. Yeung, S.-Y. R. Li, N. Cai, and Z. Zhang, “Network coding theory,” Foundations and Trends in Communications and Information Theory, vol. 2, nos.4 and 5, pp. 241–381, 2005.
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Guang, X., Zhang, Z. (2014). Coding Bounds of Linear Network Error Correction Codes. In: Linear Network Error Correction Coding. SpringerBriefs in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0588-1_4
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DOI: https://doi.org/10.1007/978-1-4939-0588-1_4
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