By Xuan Guang, Zhen Zhang (auth.)

There are major methods within the conception of community mistakes correction coding. during this SpringerBrief, the authors summarize probably the most very important contributions following the vintage strategy, which represents messages via sequences just like algebraic coding, and in addition in short talk about the most effects following the opposite technique, that makes use of the idea of rank metric codes for community errors correction of representing messages by way of subspaces. This e-book begins via constructing the elemental linear community blunders correction (LNEC) version after which characterizes similar descriptions. Distances and weights are outlined on the way to represent the discrepancy of those vectors and to degree the seriousness of mistakes. just like classical error-correcting codes, the authors additionally practice the minimal distance interpreting precept to LNEC codes at each one sink node, yet use special distances. For this interpreting precept, it truly is proven that the minimal distance of a LNEC code at each one sink node can absolutely symbolize its error-detecting, error-correcting and erasure-error-correcting services with recognize to the sink node. moreover, a few vital and important coding bounds in classical coding idea are generalized to linear community errors correction coding, together with the Hamming certain, the Gilbert-Varshamov certain and the Singleton sure. numerous confident algorithms of LNEC codes are offered, relatively for LNEC MDS codes, in addition to an research in their functionality. Random linear community mistakes correction coding is possible for noncoherent networks with blunders. Its functionality is investigated by way of estimating higher bounds on a few failure percentages via examining the data transmission and blunder correction. eventually, the fundamental concept of subspace codes is brought together with the encoding and deciphering precept in addition to the channel version, the limits on subspace codes, code building and interpreting algorithms.

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T) This is a contradiction. Therefore, we can say dmin (Cφρ ) ≥ dmin . Now, we have known that Cφρ can be regarded as a classical error-correcting (t) code with codeword length rt and the minimum distance no less than dmin , and thus we have the following inequality from the Hamming bound for classical errorcorrecting codes: qω = |Cφρ | ≤ qrt τ (t) ∑i=0 rt i (q − 1)i , and |Cφρ | = |CCUTt | = |Ct | = qω , which completes the proof. 1 is a special case of this theorem. 2 can also be regarded as the Hamming bound of linear network error correction codes.

Adding e1 , e2 , · · · , el on G and deleting e1 , e2 , · · · , el from G according to the approach mentioned in the proposition, denote by G the new network. Let CUTsρ ,t be an arbitrary cut between sρ and t in G . For any linear network code on G, let kei ,d = kei ,d , 1 ≤ i ≤ l, for each channel d ∈ E. Then for 1 ≤ i ≤ l, rowt (ei ) = rowt (ei ). 3) recursively, which implies that Δ (t, ρ ) ⊆ Δ (t,CUTsρ ,t ). Notice that it is true for any linear network code. So CUTsρ ,t dominates the error pattern ρ and rankt (ρ ) ≤ |CUTsρ ,t |.

The minimum distance decoding principle is applied to each message scalar component of the packets. 2 Distances and Weights Recall that an error pattern ρ is regarded as a set of channels in which errors occur, and an error message vector z matches an error pattern ρ , if ze = 0 for all e ∈ E\ρ . In the following, we define two vector spaces, which are of importance in further discussion. 2. Define Δ (t, ρ ) = (0 z) · F˜t : all z ∈ Z matching the error pattern ρ = {zGt : all z ∈ Z matching the error pattern ρ } , where 0 is an ω -dimensional zero row vector, and Φ (t) = (x 0)F˜t : all x ∈ F ω = {xFt : all x ∈ F ω } , where 0 is an |E|-dimensional zero row vector.