ual to Llog2 i]. The expression [x] (floor(x)) means the smallest integer near or equal to X. Thus, we have Llog 1] = 0, Llog2 2 = 1, Llog2 3 = 1, [log2 4 = 2, and SO on. According to the above rules, the Elias-gamma codewords for i = 4 and i = 13 are gamma(4) = 00_100 and gamma(13) = 000_1101, respectively (the underscore being introduced for clarity). The Elias-delta codeword is defined by gamma( (Log2 followed by the minimal binary representation of i with the most significant 1 bit being removed. With i = 4, 10.1 Integer coding for instance, we have i = 100-2 and gamma( (Log2 4] + 1) = 011, which, with this rule, yields delta(4) = 011_00. With i = 14, we have i = 1110_2 and gamma([log2 14 + 1) = gamma(4) = 00100, which, with this rule, gives delta(14) = 00100_110. Table 10.1 shows that for small integers (except i = 1) the Elias-delta codewords are longer than the gamma codewords. The lengths become equal for i = 16 to i = 31. The situation is then reversed for i > 32. This shows that Elias-delta coding is preferable for sources with i < 31, while Elias-gamma coding is preferable for larger sources. The important difference between the two codes is their asymptotic limit when the source entropy goes to infinity. It can be checked, using a short tabulating program, that the asymptotic limit of the coding efficiency, n = H/L, is equal to 50% for the Elias-gamma code, while it is equal to 100% for the Elias-delta code. 1 Therefore, the Elias-delta code is asymptotically optimal. The fact that the two Elias codes are not optimal (except asymptotically for Elias- delta) does not preclude their use for data compression. For instance, taking the English- language source with the distribution listed in Table 8.3 and the corresponding entropy H = 4.185 bit/symbol, we find that both Elias-gamma and Elias-delta codings have a mean codeword length of L = 5.241 bit/word, corresponding to an efficiency of n = H/L = 79.83%. For comparison, Huffman coding yields L = 4.212 bit/word and n = 99.33%. The Elias codes, thus, make 