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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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- Fall '13