On Minimum Distances of Network Error Control Codes

Network error control coding can be regarded as a generalization of classical algebraic coding. In particular, the minimum distance (in terms of the Hamming distance) in classical algebraic coding generalizes naturally to linear network error control coding. However, Yang and Yeung [1] surprisingly discovered that for a nonlinear network code, two different minimum distances are needed for characterizing its capabilities for error correction and error detection. In this paper, we further develop the research in this direction. Specifically, we introduce a distance and its refined version for joint error correction and detection and prove some of their basic properties, and study the relation between various distances for network error control. These results not only provide new insights into the error control capabilities of nonlinear network codes, but also enable linear network codes to be understood from a broader perspective.