L = 4.212 bit/symbol, or Hsource / L = 99.33% (with optimal codes using 3<1<91 bits); (iv) Variable-length ternary codewords with lengths between l = 1 and l = 3 trits, giving L = 2.744 trit/symbol or Hsource/L = 96.2%. The result shown in case (iii) is derived from an optimal coding approach (Huffmann coding) to be described in Chapter 9. The result shown in case (iv) will be demonstrated in the next section. At this stage, we can just observe that all the alternative solutions (i)-(iv) have coding efficiencies significantly greater than the Morse code. We also note that the efficiency seems to be greater for the multi-level (M-ary) codes with M > 2, but I will show next that it is not always true. Consider, for simplicity, the case of fixed-length M-ary codes, for which it is straight- forward to calculate the coding efficiency. For instance, quaternary codewords, made with n quad characters, called 0, 1, 2, and 3, can generate 4n symbol possibilities. Since 42 < 26 < 4 superscript(3) = 64, codewords with 3 quads are required for the A-Z alpha- bet. The mean codeword length is, thus, L = 3 quad/symbol. The source entropy is H4 = (In 2/In 4)H2 = 2.092 quad/symbol. Thus, the coding efficiency is n = H4/L = 69.7%, which is lower than that of ternary, fixed-length coding (88.0%), as seen in case (ii). The reason is that going from ternary to quaternary coding does not reduce the codeword length, which remains equal to three. The situation would be quite different Information coding if we used variable-length codewords, since a single quad can represent four possibil- ities, instead of three for the trit. Let us look at the extreme case of a 26-ary coding, i.e., a coding made of 26 alphabet characters (called 26-its) which can be represented, for instance, by an electrical current with 26 intensity levels. Each codeword is, thus, made of a single 26-it, and the mean codeword length is L = 1 26-it/symbol. The source entropy is H26 = (ln2/1n26)H2 = 0.890 26-it/symbol. The coding efficiency is, there- fore, n = H26/L =