A Revisit of Linear Network Error Correction Coding

We consider linear network erro correction (LNEC) coding when errors may occur on the edges of a communication network of which the topology is known. In this paper, we first present a framework of additive adversarial network for LNEC coding, and then prove the equivalence of two well-known LNEC coding approaches, which can be unified under this framework. Furthermore, by developing a graph-theoretic approach, we obtain a significantly enhanced characterization of the error correction capability of LNEC codes in terms of the minimum distances at the sink nodes. Specifically, in order to ensure that an LNEC code can correct up to
$r$
errors at a sink node
$t$
, it suffices to ensure that this code can correct every error vector in a reduced set of error vectors; and on the other hand, this LNEC code in fact can correct every error vector in an enlarged set of error vectors. In general, the size of this reduced set is considerably smaller than the number of error vectors with Hamming weight not larger than
$r$
, and the size of this enlarged set is considerably larger than the same number. This result has the important implication that the computational complexities for decoding and for code construction can be significantly reduced.