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CHAP TER 10 10.1 INTRODUCING CODES DEFINED BY A SET OF CONSTRAINTS In Ch. 6, we deﬁned an error-correcting code C(n, k) as a subset among the set S of n-symbol sequences over some ﬁnite alphabet of size q. The number of symbols k of the information message that each codeword represents is referred to as the number of dimensions of the code or, more brieﬂy, as its dimension. The code being necessarily redundant implies the strict inequality k< n. There are basically two ways for deﬁning such a code. One way is to give the list of its words; random coding, as used by Shannon and his followers in order to prove the fundamental theorem of channel coding, is a typical example of it. The trouble with a mere list of codewords is that the number of its elements exponentially grows in terms of the codeword length n: assuming a k Rn R n constant information rate R = k/n, 0 <R < 1, the number of codewords is q = q = (q ) , which makes the decoding process of such a code of prohibitive complexity when n is large enough for the code to be
Published: Jan 1, 2008
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