By Stuart E. Dreyfus

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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.