# 723 mechanics of division there are several efcient

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Unformatted text preview: ihood of the message being correct, as long as a suitable generator polynomial is used. ￿ 7.2.3 Mechanics of division There are several efﬁcient ways to implement the division and remaindering operations needed in a CRC computation. The schemes used in practice essentially mimic the “long division” strategies one learns in elementary school. Figure 7-1 shows an example to refresh your memory! SECTION 7.2. CYCLIC REDUNDANCY CHECK ￿ 7.2.4 5 Good Generator Polynomials So how should one pick good generator polynomials? There is no magic prescription here, but what commonly occuring error patterns do to the received code words, we can form some guidelines. To develop suitable properties for g (x), ﬁrst observe that if the receiver gets a bit sequence, we can think of it as the code word sent added to a sequence of zero or more errors. That is, take the bits obtained by the receiver and construct a polynomial, r(x) from it. We can think of r(x) as being the sum of w(x), which is what the sender sent (the receiver doesn’t know what the real w was) and an error polynomial, e(x). Here’s the key point: If r(x) = w(x) + e(x) is not a multiple of g (x), then the receiver is guaranteed to detect the error. Because w(x) is constructed as a multiple of g (x), this statement is the same as saying that if e(x) is not a multiple of g (x), the receiver is guaranteed to detect the error. On the other hand, if r(x), and therefore e(x), is a multiple of g (x), then we either have no errors, or we ha...
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