L6_2

# By linear we mean that any given bit in a valid code

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Unformatted text preview: y” bits). Often, we use the notation (n, k, d), where d refers to the minimum Hamming distance of the block code. The rate of a block code is deﬁned as k/n; the larger the rate, the less the overhead incurred by the code. A linear code (whether a block code or not) produces code words from message bits by restricting the algebraic operations to linear functions over the message bits. By linear, we mean that any given bit in a valid code word is computed as the weighted sum of one or more original message bits. Linear codes, as we will see, are both powerful and efﬁcient to implement. They are widely used in practice. In fact, all the codes we will study—including convolutional codes—are linear, as are most of the codes widely used in practice. We already looked at the properties of a simple linear block code: the replication code we discussed in Section 6.2 is a linear block code with parameters (c, 1, c). To develop a little bit of intuition about the linear operations, let’s start with a “hat” puzzle, which might at ﬁrst seem unrelated to coding. There are N people in a room, eac...
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