We now turn to developing more sophisticated codes

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Unformatted text preview: STANCE 5 the replication code is extremely inefficient in terms of the overhead it incurs. As such, it is used only in situations when bandwidth is plentiful and there isn’t much computation time to implement a more complex decoder. We now turn to developing more sophisticated codes. There are two big ideas: embedding messages into spaces in a way that achieves structural separation and parity (linear) computations over the message bits. ￿ 6.3 Embeddings and Hamming Distance Let’s start our investigation into error correction by examining the situations in which error detection and correction are possible. For simplicity, we will focus on single error correction (SEC) here. There are 2n possible n-bit strings. Let’s define the Hamming distance (HD) between two n-bit words, w1 and w2 , as the number of bit positions in which the messages differ. Thus 0 ≤ HD(w1 , w2 ) ≤ n. Suppose that HD(w1 , w2 ) = 1. Consider what happens if we transmit w1 and there’s a single bit error that inconveniently occurs at the one bit position in which w1 and w2 differ. From the receiver’s point of view it just received w2 —t...
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