takes L = 1.311 bit/word to describe any game sequence uniquely, corresponding to a coding efficiency of n = 53.544%. The efficiency of our block code (n = 53.544%) is not outstanding, indeed, but we can measure the progress made by comparing it to that given by a fixed-length code. Indeed, if we use a four-bit codeword to describe any of the game sequences (e.g., 1010 for WLWL), the efficiency drops to n = 4H/4 = 0.1755 or 17.5%. The use of our variable-length block code has, in fact, more than doubled the coding efficiency! With respect to four-bit codewords, the compression rate obtained with the block code is r = 1 - 1.311/4 = 67.2%. This first example with sequences of n = 4 games illustrates that the efficiency of the block code increases with the length of the sequence. Indeed, long sequences of events contain more information than short sequences of events, because their associated prob- abilities are lower.23 Therefore, we can infer that our block coding becomes increasingly efficient with longer sequences, which we shall now verify. Table 9.5 shows the results obtained with sequences of length n = 32. As the table first shows, the number of events is dramatically increased to a whopping 4294967296 or about 4.3 billion possibilities! The code is seen to have an average length of L = 5.771 bit/word, corresponding to a coding efficiency of n = 97.34%. The corresponding compression rate is r = 1 5.771/32 = 81.9%. The results obtained for n = 32 games seem to confirm the previous observation, according to which the code becomes increasingly efficient with ever-longer sequences. As it turns out, however, this is not the case! Figure 9.8 shows a plot of the coding efficiency N, as computed with sequence lengths from n = 1 to n = 129. From the figure, we observe that the efficiency increase exhibits some irregularities, with local peaks appearing whenever n is a power of two (n = 2k), but also for other intermediate values. The irregularity observed between the peaks is explained by the fact that the trailer bi