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In this paper, we study basic properties of linear network error correction codes, their construction and error correction capability for various kinds of errors. Our discussion is restricted to the single-source multicast case. We define the minimum distance of a network error correction code. This plays the same role as it does in classical coding theory. We construct codes that can correct errors up to the full error correction capability specified by Singleton bound for network error correction codes recently established by Cai and Yeung. We propose a decoding principle for network error correction codes, based on which we introduce two decoding algorithms and analyze their performance. We formulate the global kernel error correction problem and characterize the error correction capability of codes for this kind of error.