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Figure 4.12. Scheme illustrating the encoding process.
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With much detail the encoding process for a single outgoing packet X works as follows: x0 = mi 0 0 m1 0 = g0 m2 ... ... m0 0 m1 1 1 m + g1 2 ... ... m1 m0 l 1 l 1 m1 0
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x 1 x2 X= ... ... xl 1
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mi 1 i n 1 n 1 m gi M i = gi 2 ... i=0 i=0 ... mi l 1 m2 1 2 m +g2 2 ... ... m2 0
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where xi , mi , gj GF (2s ) xi is the ith symbol of the egress vector (packet) X. mi is the ith symbol of the jth ingress packet M j . gj is the coef cient that multiplies the jth ingress packet M j . The same encoding process is then repeated more times. Speci cally, from n ingress packets a node generates m outgoing, egress packets with m n so that at the receiving site decoding is possible after having received any n out of m packets. Hence, the encoding node actually generates m sets of coef cients, one for each egress packet to generate, and then performs encoding as follows (all the three formulas below are equivalent): Xj = X0
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where Xi is the ith egress packet that is generated and gi is the ith symbol of the jth encoding vector that is, the jth set of coef cients which is generated to produce the j jth encoded packet. All the cof cients gi together form the (m n)-sized encoding j is named information vector, and the corrematrix G. Each encoded data packet X sponding vector of coef cients it is obtained from, gj , is said to be an encoding vector. The encoding node produces and sends out packets by including both encoding and information vectors; thus an outgoing packet is actually a tuple < gj , Xj > for j = 0, . . . , m 1. Each encoding vector includes n symbols, whereas each information vector includes l symbols, as many as in the ingress data items M i , i = 0, . . . , n 1: gj , Xj = (g0 , g1 , g2 , . . ., gn 1 ), (x0 , x1 , x2 , . . ., xl 1 ) When an encoding vector includes all zeros but one single one in the ith position, it means that the ith ingress data item is not encoded. gi = ei = (0, 0, . . ., 0, 1, 0, . . ., 0) Xi = M i Recursive Encoding. Suppose that the following conditions hold. 1. An intermediate relay receives and stores a set of r-tuples, as follows: g0 , X0 , g1 , X1 , g2 , X2 , . . ., gr 1 , Xr 1
j j j j j j j j
NETWORK CODING
2. All the information vectors received (X0 , X1 , X2 , . . . , Xr 1 ) come from the same set of original data (M0 , M1 , M2 , . . . , Mn 1 ) produced by a single source node somewhere. Then, the intermediate node performs re-encoding on the set of information vectors n because that it has received: X0 , X1 , X2 , . . ., Xr 1 . It is not necessary that r the intermediate node does not need decoding at this stage. It simply re-encodes the pieces of data it has received. This results in adding some more redundancy into the network for some part of the original data. Re-encoding is performed by generating a certain number of encoding vectors, as many as the number of information vectors that the node wants to send out. Suppose, for example, that the node wants to generate k information vectors. It should be noted that there is not a mathematical constraint on the values allowed for r and k. Rather, the right tuning of these parameters depends on the networking scenario and is still a challenging open issue. After having chosen the appropriate value for k, the relay node acts as follows: a. It selects k new encoding vectors hj , j = 0,. . . , k 1. Each encoding vector includes as many symbols as the number of ingress information vectors to which the new encoding is applied. In this case each encoding vector is r symbols long:
hj = (h0 , h1 , h2 , . . ., hr 1 ),