By David R. Finston and Patrick J. Morandi

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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. However, we do not yet know i, so we cannot yet determine v from w. However, HwT = H(v + ei )T = Hv T + HeTi = HeTi ; and HeTi is the i-th column of H, as we pointed out above.

We see that the distance of C is 3, so C is 1-error correcting. 4. COSET DECODING 39 Syndrome 000 101 010 011 100 110 111 001 Coset Leader 00000 10010 00010 00100 01000 01010 10000 00001 To see how we use the syndrome table to decode, we give an example. Suppose that w = 10010 is received. If we calculate Hw, we get 101. First of all, since Hw 6= 0, the vector w is not a codeword. By looking at the syndrome table, 101 is the second syndrome listed. The corresponding coset leader is e = 10010. We then decode w as c = w + e = 00000.

Let C = f00000; 00111; 11100; 11011g. The distance of C is 3, and so C is a 1-error correcting code. 9. Let n be an odd positive integer, and let C = f0 0; 1 1g be a code of length n. If n = 2t + 1, then C is a t-error correcting code since the distance of C is n. Thus, by making the length of C long enough, we can correct any number of errors that we wish. However, note that the fraction of components of a word that can be corrected is t=n, and this is always less than 1=2. Exercises 1. Find distance and error correction capability of the following codes: (a) f0000000; 1010101; 0101010; 1111111g, (b) f00000000; 11111111; 11100000; 00011111g, (c) f00000000; 11110000; 00001111; 10101010; 11111111; 01011010; 10100101; 01010101g.