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I Theorem 31.2 Correcting Capability of a Linear Code If the Hamming weight of a linear code is at least 2t + 1, then the code can correct any t or fewer errors. Alternatively, the same code can detect any 2t or fewer errors. PROOF We will use nearest-neighbor decoding; that is, for any received vector v, we will assume that the corresponding code word sent is a code word v' such that the Hamming distance d(v, v') is a minimum. (If there is more than one such v', we do not decode.) Now, suppose that a transmitted code word u is received as the vector v and that at most t er- rors have been made in transmission. Then, by the definition of distance between u and v, we have d(u, v) < t. If w is any code word other than u, then w – u is a nonzero code word. Thus, by assumption, 2t + 1< wt(w – u) = d(w, u) < d(w, v) + d(v, u) < d(w, v) + t, and it follows that t + 1< d(w, v). So, the code word closest to the re- ceived vector v is u, and therefore v is correctly decoded as u. To show that the code can detect 2t errors, we suppose that a trans- mitted code word u is received as the vector v and that at least one error, but no more than 2t errors, was made in transmission. Because only code words are transmitted, an error will be detected whenever a received word is not a code word. But v cannot be a code word, since d(v, u) < 2t, whereas we know that the minimum distance between distinct code words is at least 2t + 1.

I Theorem 31.2 Correcting Capability of a Linear Code If the Hamming weight of a linear code is at least 2t + 1, then the code can correct any t or fewer errors. Alternatively, the same code can detect any 2t or fewer errors. PROOF We will use nearest-neighbor decoding; that is, for any received vector v, we will assume that the corresponding code word sent is a code word v' such that the Hamming distance d(v, v') is a minimum. (If there is more than one such v', we do not decode.) Now, suppose that a transmitted code word u is received as the vector v and that at most t er- rors have been made in transmission. Then, by the definition of distance between u and v, we have d(u, v) < t. If w is any code word other than u, then w – u is a nonzero code word. Thus, by assumption, 2t + 1< wt(w – u) = d(w, u) < d(w, v) + d(v, u) < d(w, v) + t, and it follows that t + 1< d(w, v). So, the code word closest to the re- ceived vector v is u, and therefore v is correctly decoded as u. To show that the code can detect 2t errors, we suppose that a trans- mitted code word u is received as the vector v and that at least one error, but no more than 2t errors, was made in transmission. Because only code words are transmitted, an error will be detected whenever a received word is not a code word. But v cannot be a code word, since d(v, u) < 2t, whereas we know that the minimum distance between distinct code words is at least 2t + 1.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.2: Direct Methods For Solving Linear Systems
Problem 3CEXP
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. Prove that for nearest-neighbor decoding, the converse of Theorem
31.2 is true.

I Theorem 31.2 Correcting Capability of a Linear Code If the Hamming weight of a linear code is at least 2t + 1, then the code can correct any t or fewer errors. Alternatively, the same code can detect any 2t or fewer errors. PROOF We will use nearest-neighbor decoding; that is, for any received vector v, we will assume that the corresponding code word sent is a code word v' such that the Hamming distance d(v, v') is a minimum. (If there is more than one such v', we do not decode.) Now, suppose that a transmitted code word u is received as the vector v and that at most t er- rors have been made in transmission. Then, by the definition of distance between u and v, we have d(u, v) < t. If w is any code word other than u, then w – u is a nonzero code word. Thus, by assumption, 2t + 1< wt(w – u) = d(w, u) < d(w, v) + d(v, u) < d(w, v) + t, and it follows that t + 1< d(w, v). So, the code word closest to the re- ceived vector v is u, and therefore v is correctly decoded as u. To show that the code can detect 2t errors, we suppose that a trans- mitted code word u is received as the vector v and that at least one error, but no more than 2t errors, was made in transmission. Because only code words are transmitted, an error will be detected whenever a received word is not a code word. But v cannot be a code word, since d(v, u) < 2t, whereas we know that the minimum distance between distinct code words is at least 2t + 1.
Transcribed Image Text:I Theorem 31.2 Correcting Capability of a Linear Code If the Hamming weight of a linear code is at least 2t + 1, then the code can correct any t or fewer errors. Alternatively, the same code can detect any 2t or fewer errors. PROOF We will use nearest-neighbor decoding; that is, for any received vector v, we will assume that the corresponding code word sent is a code word v' such that the Hamming distance d(v, v') is a minimum. (If there is more than one such v', we do not decode.) Now, suppose that a transmitted code word u is received as the vector v and that at most t er- rors have been made in transmission. Then, by the definition of distance between u and v, we have d(u, v) < t. If w is any code word other than u, then w – u is a nonzero code word. Thus, by assumption, 2t + 1< wt(w – u) = d(w, u) < d(w, v) + d(v, u) < d(w, v) + t, and it follows that t + 1< d(w, v). So, the code word closest to the re- ceived vector v is u, and therefore v is correctly decoded as u. To show that the code can detect 2t errors, we suppose that a trans- mitted code word u is received as the vector v and that at least one error, but no more than 2t errors, was made in transmission. Because only code words are transmitted, an error will be detected whenever a received word is not a code word. But v cannot be a code word, since d(v, u) < 2t, whereas we know that the minimum distance between distinct code words is at least 2t + 1.
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