Linear Encoding

In Section 4.6.1.2 we showed the achievability of the capacity region as defined in Theorem 4.2 by using general random coding functions at different nodes of the network. In this section we restrict our attention to linear encoding schemes. The advantage of using a linear encoding scheme is that the decoding process becomes much easier. In this case, the equivalent transfer function of the network from any source to any destination, having the erasure locations at that destination, is linear. Hence, decoding at the destination is simply forming and solving a linear system of

equations.

In this section we show that linear encoders achieve capacity. Let us first define the linear block coding scheme with block length ofn:

Recall that W(s) _{= {1,2, . . . ,d2}nRs_{e} is the message set for information source}

s_{∈ S}. We assume that different messages are equiprobable and independent of each other. For anyw(s) _{∈ W}(s)_{, let b(w}(s)_{) be the length-nR}

sbinary expansion ofw(s)−1. The encoding operation is as follows:

Each node i ∈ V transmits n linear combinations of the non-erased symbols re- ceived from its incoming edges and the binary representation of the message it wants to transmit across the network. More precisely, node i generates a random binary matrixBi of size n×n(dI(i) +Ri) wheredI(i) is the in-degree of nodeiand Ri is the rate of the codebook used at nodei(in the case whereiis not a source of information

Ri = 0). Each element of Bi is drawn i.i.d. Bernoulli(1/2). For a given sequence

y, let _ey be a sequence derived by replacing every e with 0. Note that _ey and y have the same length.12 _{If node i receives Y}n

i = yin on its incoming edges and wants to transmit messagew(i)_{then it sends outx}

i =Bi·[b(w(i)),yein]†. (Since the input-output relation at each node is linear, setting the erased symbols equal to zero is the same as finding linear combinations of only the non-erased bits.)

Each destination d knows all the matrices Bi and also the erasure locations Zn on all the links across the network. Since each received and transmitted symbol at any node is a linear combination of the elements of vector b(w) = (4 b(w(s)_{), s ∈ S).}

Therefore each destination receives a collection of linear combinations of elements of b(w). Using _{Bi}i∈V and Zn, destination node d can construct the matrix that

corresponds to the linear input-output relation of the network. We denote this matrix byF(_{Bi}, Zn), givingYf_dn(w) =F({Bi}, Zn)·b(w)†. Note that matrixF is a function of different nodes’ encoding matrices {Bi} and Zn.

Now, upon receiving Yn

d =y∈ {0,1, e}ndI(d), the destination nodedlooks (solves) for the message vector w ∈ W =4 Q_s∈SW(s) _{such that F({M}

i}, Zn)·b(w)† = ye. If

12_{The corresponding mapping from alphabet GF(q)∪ {e}to GF(q) again replacesewith 0. This} variation is useful for packet erasure networks.

there is a unique wwith this property, node ddeclares it as the transmitted message vector, otherwise it declares an error. Note that the actual transmitted message vector, say w_{0 ∈ W}, always satisfies the above property. Therefore, an error occurs only if there is another message vector w6=w₀ such that Yn

d (w) =Ydn(w0) =y.

4.7.1

Achievability Result for Linear Encoding

Looking at the achievablity proof and probability of error analysis for general random coding in Sections 4.6.1.2 and 4.6.1.3, it can be easily verified that the linear case requires the same error events (4.10). Since the erasure vector Zn _{is available at the} destination, there is no difference between Yei and Yi and we can determine one from the other. By expanding the conditional error event E(w) given A(_δn) for different cuts in the network, all of the relations up to step (d) of equation (4.14) go through for the linear case. In fact the relations up to step (d) only require the independence of encoding functions for different nodes of the network, which holds for the linear case. Now we look at the following probability in (4.14)

Pi = Pr4 _\ j:(i,j)∈[Vx,Vc x] {Y_ijn(w) =Y_ijn(w0)} A(δn), {(w(i), Yin(w))6= (w (i) 0 , Yin(w0))} . (4.18) As in the general random coding argument, for a fixed i we have Yn

ij(w) = Yijn(w0)

for all j such that (i, j)_∈[_Vx,Vxc], only if Xin(w) and Xin(w0) differ only in locations

where an erasure occurs on all the edges of the interest. Because of strong typicality, the number of these location is at most n(Q_{j: (i,j)∈[Vx,V}c

x]ij+δ). Therefore X n i (w) and Xn

i (w0) should be the same in at least n(1−

Q

j: (i,j)∈[Vs,Vc

s]ij −δ) locations. But by our encoding scheme this means that

Bi·([w(i), Yin(w)]†−[w

(i)

0 , Yin(w0)]†)

| {z }

z should be zero in at least n(1₋Q_{j: (i,j)∈[Vs,V}c

s]ij+δ) specific locations. Also note that since (w(i)_{, Y}n

i (w)) 6= (w

(i)

argument we have

Pi ≤ Pr

Bi·zbe 0 in at least nαi specific locations

z 6= 0 (a) ≤ 2−nαi = 2−n(1−Qj: (i,j)∈[Vx,Vxc]ij−δ), (4.19) whereαi = 1−Q_{j: (i,j)∈[Vx,V}c

x]ij−δ and (a) follows from the following lemma and its corollary. Proof of this lemma is provided in Appendix 4.9.2.

Lemma 4.6. Let X be a non-zero vector of size n_×1 from some finite field GF(q).

Suppose that A is a random matrix of size m×n with i.i.d. components distributed

uniformly over GF(q). Then the coordinates of Y =A_·X are i.i.d. uniform random

variables over GF(q).

Corollary 4.7. The probability that Y =A_·X is zero ink specific coordinates equals

q−k_.

Now note that by replacing Pi in (4.18) and (4.14) with its bound from (4.19) we get the same bound as (4.16) for random linear codes. Therefore linear operations are sufficient for achieving the capacity.

The main advantage of this is that the decoding operation can be carried out without exhaustive search of the exponential-sized codebook. The destination(s) only has to solve a system of linear equations, which can be done in polynomial time. This allows for faster and more efficient network operation.

4.8

Conclusions

We have obtained the capacity for a class of wireless erasure networks with broadcast and no interference at reception. We have generalized some of the capacity results that hold for wireline networks [4, 58] to these networks. Furthermore, we have shown that linear encoding suffices to achieve the optimal performance. We see from the proof that it is not necessary to perform channel coding and network coding separately from each other. In fact in [19, 62] we show that decoding at the relay nodes and operating

below the capacities of each link can actually significantly reduce the achievable rate. Therefore, unlike the wireline scenario where each link is made error-free by channel coding, and network coding is then employed on top of that, our scheme only requires a single encoding function. Only the destination has to decode the received signal.

Many problems related to wireless networks remain open. Generalizing the results in this chapter for other network problems is one possible extension of this work. As a first step, in [2], the problem of a single source wanting to send the same information to several destinations is considered. For these problems it can be shown that unlike wireline networks, the capacity region is not given by min-cut bounds. It is shown in [24] that the capacity region of multiple input erasure broadcast channels is given by time-sharing between users at different inputs. This result gives tighter outer-bounds on the capacity region of broadcast problems in erasure wireless networks.

It will also be interesting to see if similar results can be obtained for other types of networks, such as erasure wireless networks in which interference is incorporated in the reception model, networks involving channels other than erasure channels, etc.