It is well known that the minimum distance for linear network codes plays the same role as the minimum distance for classical error control codes. However, Yang and Yeung (2008) discovered that for nonlinear network codes, the minimum distance for error correction is not always the same as the minimum distance for error detection. Inspired by the idea that the channel will affect the distances between the codewords, we establish the scheme of a generalized network channel and a generalized network code. Then, we systematically define the distances for error correction and error detection under the scheme of the generalized network code. We consider the joint error correction and detection in the generalized network code and obtain a complete characterization by introducing a distance and its refined version for this purpose. We enhance our understanding of the relation between various distances for error correction and detection in generalized network codes by proving some bounds on these distances.

翻译：众所周知，线性网络编码的最小距离在纠错编码中扮演着与经典纠错码最小距离相同的角色。然而，杨和杨英卓（2008）发现，对于非线性网络编码，纠错最小距离与检错最小距离并不总是相同。受信道会影响码字间距离这一思路的启发，我们建立了广义网络信道与广义网络编码的框架。在此基础上，我们系统地定义了广义网络编码框架下用于纠错和检错的距离。我们考虑了广义网络编码中的联合纠错与检错，并通过引入一种距离及其精细化版本，获得了完整的刻画。通过证明这些距离上的若干界，我们加深了对广义网络编码中纠错与检错各种距离之间关系的理解。