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nodes and assume that all channels have capacity 1. This paper summarizes our recent works on network error Let T be a subsets of V whose elements are called sink correction codes. We study basic properties of linear network nodes. Let s C V be the source node. The rest of the nodes in error correction codes in the single source multicast case [8]. I = V -S{s} -T are called internal nodes. Let F be a finite We define the minimum distance of a network error correction field of sufficiently large cardinality. The source messages are code which plays the same role as it does in classical coding w packets X = (Xi: 1i = 1, w) where Vti,Xi FK theory. We construct MDS codes and give sufficient conditions and K is the packet length. Let Xi = (Xil,... , XiK) where for its existence. We propose basic decoding algorithms and for all j:1. j < K, Xij C F. They are transmitted to analyze their performance. We propose an improved upper the source node s through w imaginary channels in In(s). bound for the failure probability of random network code and At each node i e V -T, there is a local encoding kernel use it to analyze the performance of randomized network error matrix Ki = (kde: d C In(i), e C Out(i)) where kde C .F. correction codes [9], [10]. We study the possibility of decoding At source node s, we assume that the packet transmitted over beyond error correction capability [11]. We propose a hybrid the ith imaginary channel di is the source message packet network error correction coding systems. An extensive perfor-Ud, = Xi. The packet transmitted over channel e (i,j), mance analysis of this coding system is reported in a separate denoted by Ue, is calculated by the following formula paper [12]. Ue S kdeUd. (1) deln(i)

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