Download PDF by David R. Finston and Patrick J. Morandi: An Introduction to Abstract Algebra via Applications

By David R. Finston and Patrick J. Morandi

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Then a b = ab = n = 0. Since a 6= 0, and b 6= 0 we …nd that both are zero divisors. But a …eld has no zero divisors, so Zn is not a …eld.. 28. Let F = f0; 1; a; bg be a set with four elements, and de…ne operations by the following tables + 0 1 a b 0 1 0 1 1 0 a b b a a a b 0 1 b b a 1 0 0 1 a b 0 1 0 0 0 1 0 a 0 b a 0 a b 1 b 0 b 1 a : A painful and tedious computation will show that F , together with these operations, is a …eld, which we will not bother with. This example looks very ad-hoc. However, there is a systematic way to obtain this example from the …eld Z2 ; we will need to use this method of building …elds from others to work with 2 Reed-Solomon codes, an important class of error correcting codes.

If wt(s) > 3 and wt(s + bi ) 2 for some i, then v = w + (s + bi ; ei ). 4. If we haven’t yet determined v, then compute Bs. 5. THE GOLAY CODE 5. If 1 wt(Bs) 41 3, then v = w + (0; Bs). 6. If wt(Bs) > 3 and wt(Bs + bi ) 2 for some i, then v = w + (ei ; Bs + bi ). 7. If we haven’t determined v, then w cannot be decoded. 22. Suppose that w = 001001001101101000101000 is received. We calculate s = Hw, and we get s = 110001001001. We see that wt(s) = 5. We see that wt(s + b5 ) = 2. Therefore, by Step 3, we decode w as v = w + (s + b5 ; e5 ) = w + (000000010010; 000010000000) = 001001011111101010101000.

The code C has a particularly elegant decoding algorithm, which we now describe. Let e1 ; : : : ; e7 be the standard basis for Z72 . We point out a simple fact of matrix multiplication: 36 CHAPTER 2. ERROR CORRECTING CODES HeTi is equal to the i-th column of H. Moreover, we note that the 7 nonzero vectors in K 3 are exactly the 7 columns of H. Suppose that v is a codeword that is transmitted as a word w 6= v. Suppose that exactly one error has been made in transmission. Then w = v + ei for some i.

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An Introduction to Abstract Algebra via Applications by David R. Finston and Patrick J. Morandi

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