By Gareth A. Jones
This article is an uncomplicated advent to info and coding thought. the 1st half specializes in info conception, overlaying uniquely decodable and on the spot codes, Huffman coding, entropy, info channels, and Shannon’s basic Theorem. within the moment half, linear algebra is used to build examples of such codes, corresponding to the Hamming, Hadamard, Golay and Reed-Muller codes. includes proofs, labored examples, and routines.
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2, 1: S -t S' -t . . -t S(q-2) -t S(q-l) . Now S(q-l) has a single symbol 81 V . . V 8 q of probability 1, and we use the empty word e to encode this, giving a code! C(q-l) = {c} for S(q-l) . The above process of adding 0 and 1 to a code-word Wi then gives us an instantaneous binary code C(q-2) = {cO = 0, s l = I} for S(q-2), and by repeating this process q - 1 times we get a sequence of binary codes C(q-l) , C(q-2) , . . , C I, C for the sources S(q -l), S(q-2), . . , S', S: -t SI -t -t C +-- CI +-- +-- C(q-2) +-- C(q-l) .
Xi 1. When some Xi 0 the argument is similar, since our convention that Xi 10g(1/xd = 0 allows us to ignore such terms. 10 If a source S has q symbols then Hr(S) ~ log, q, with equality if and only if the symbols are equiprobable. 9 are satisfied. We therefore have q 1 q q Hr(S) = LPi log, -:- ~ LPi log, q = log, q LPi = log, q, i=1 PI i=1 i=1 with equality if and only if each Pi = l/q. [J Thus the entropy is greatest, and the most information is conveyed, when there is the greatest uncertainty about the symbols emitted.
6 Let Shave q = 5 symbols 81, .. ,85 again, but now suppose that they are equiprobable, that is, P1 = ... 2. 4 1 00 1 1 0 00 e This gives a Huffman code C = {01, 10, 1l,000,001} for S; it is a prefix code, and is therefore instantaneous. 4. 5, where the symbols s, were not equiprobable. In general, the greater the variation among the probabilities Pi, the lower the average word-length of an optimal code, because there is greater scope for assigning shorter code-words to more frequent symbols. We will study this phenomenon more systematically in Chapter 3, using a concept called entropy to measure the amount of variation in a probability distribution.