e length n, it is straightforward to tabulate the suite of n+1 eigen- values according to the definition Xa-12b Na-226 (22.38) 2225-2 The parameter E(n,p) corresponds to each eigenvalue Mi (n,p) = as defined (22.39) Figure 22.1 shows the values of the parameter E(n,p) in the range n = 1 to n = 20. The values are connected by full lines to guide the eye. The top and bottom horizontal lines correspond to E(n, n) and E(n,0) = E(Mi = Xa), respectively. The two sets circled at n = 3 and n = 19 help visualize the effect according to which an increasing 22.3 A graphical and numerical illustration Message length Figure 22.2 Lower bound for the parameter E(n, p), calculated from Eq. (22.39), with zoom in the region 0.4. fraction of the E(n, p) values progressively form a cluster in the region between E = 0 and E(n,0) < 0.4. This effect is also illustrated in Fig. 22.2, which shows a zoom in this region. It is clear that for a message length n sufficiently large, any value E > 0 can be reached by E(n, p). Furthermore, we observe that given a message length n and any > 0 it is always possible to define a subset of codewords satisfying E(n, p) < E, as indicated in the figure by the circled clusters. In the case n = 19, for instance, the cluster consists of a subset of five codewords satisfying E (19, p) < 0.29681, and this group of codewords is precisely the E-typical subspace So we have been looking for! Let us take a closer look at the corresponding data. Table 22.4 shows, for each codeword type u of length n = 19, from left to right: the individual codeword probability p(u), the number of possible permutations N of the codeword type u, the codeword-type probability Np(u), the codeword parameter E, and the associated value R = S+E. We have defined our E-typical subspace S as generated by the five codeword types for which E(19, 0.29681. As the table shows, these codeword types are of the form 12.1623), 121526), and 121425). Summing the Ndata, it can be found that there exist 16 663 such codewords, out of 524 288 possibiliti