By Gareth A. Jones

This article is an straightforward advent to details and coding thought. the 1st half specializes in details thought, overlaying uniquely decodable and on the spot codes, Huffman coding, entropy, details channels, and Shannon’s primary Theorem. within the moment half, linear algebra is used to build examples of such codes, comparable to the Hamming, Hadamard, Golay and Reed-Muller codes. comprises proofs, labored examples, and workouts.

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1 Let S be an unbiased coin, with 81 and 82 representing heads and tails . Then = P2 = ~, so if we take r = 2 then h(8d = 12(82) = 1. Thus the standard unit of information is how much we learn from a single toss of an unbiased coin. P1 Since each symbol s, of a source S is emitted with probability Pi, it follows that the average amount of information conveyed by S (per source-symbol) is given by the function called the r-ary entropy of S. As with the function 1, a change in the base r corresponds to a change of units, given by When r is understood, or unimportant, we will simply write H(S) q 1 q i=l P, i=l = I>i log -:- = - LPi logpi .

Proof Independence gives Pr(sitj) Hr (8 x T) =- L = Piqj, so ~:::>iqj 10grPiqj i j =- L LPiqj(logrPi i + log, qj) j =- L L Piqj Iog, Pi - L L Piqj log, qj i j i j = (- L Pi log, Pi) (L qj) + (L Pi) ( i since l: Pi = l: qj = 1. j i L qj log, qj) j o 48 Information and Coding Theory We can use induction to extend the definition of a produc t to any finite number of sources: we define St x ... X S« = (St X ... X Sn-t ) The sources S, are independent if each symbol where each Si; has probability P i; .

If we continue to reduce sources in this way, we obtain a sequence of sources S , SI , .. , S(q -2), S(q -l) with the number of symbols successively equal to q, q - 1, .. , 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) , .