Construction of Network Error Correction Codes in Packet Networks

• Published in: IEEE transactions on information theory Vol. 59; no. 2; pp. 1030 - 1047
• Main Authors: Guang, Xuan, Fu, Fang-Wei, Zhang, Zhen
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Coding theory; Coding, codes; Communications networks; Decoding; Encoding; Error correction & detection; Error correction codes; Exact sciences and technology; Extended global encoding kernels; Information theory; Information, signal and communications theory; Kernel; maximum distance separable (MDS) code; Network coding; network error correction code construction; network error correction coding (NEC); Probability; random linear network error correction coding; refined Singleton bound; Signal and communications theory; Switching and signalling; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Vectors; Performance evaluation; Packet switched network; Packet switching; refined Singleton bound; maximum distance separable (MDS) code; network error correction coding (NEC); Random error; Random coding; Algorithm; Failures; Kernel method; Degradation; Upper bound; Linear coding; Extended global encoding kernels; network error correction code construction; random linear network error correction coding; Telecommunication network; Network coding; Error correcting code; Minimal distance; Error correction; Damaging
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2012.2222344

Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to NEC, especially the Singleton bound. In this paper, following the research line using the extended global encoding kernels proposed by Zhang in 2008, the refined Singleton bound of NEC can be proved more explicitly. Moreover, we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable (MDS) codes can become smaller further. By this proof, an algorithm is proposed to construct general linear network error correction codes including the linear network error correction MDS codes. Finally, we study the error correction capability of random linear NEC. Motivated partly by the performance analysis of random linear network coding, we evaluate the different failure probabilities defined in this paper in order to analyze the performance of random linear NEC. Several upper bounds on these probabilities are obtained and they show that these probabilities will approach to zero as the size of the base field goes to infinity. Using these upper bounds, we slightly improve on the probability mass function of the minimum distance of random linear network error correction codes in a paper by Balli and colleagues, as well as the upper bound on the field size required for the existence of linear network error correction codes with degradation at most d .