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Unformatted text preview: and instructive exercise for the reader.
The rest of this section shows how to construct linear block codes over F2 . For simplicity, and without much loss of generality, we will focus on correcting single-bit errors.
We will show two ways of building the set S of transmission messages such that the size
of S will allow us to send messages of some speciﬁc length, and to describe how the receiver can perform error correction on the (possibly corrupted) received messages. These
are both examples of single error correcting (SEC) codes.
We will start with a simple rectangular parity code, then discuss the cleverer and more
efﬁcient Hamming code in Section 6.4.3.
6.4.1 Rectangular Parity SEC Code Let parity(w) equal the sum over F2 of all the bits in word w. We’ll use · to indicate the
concatenation (sequential joining) of two messages or a message and a bit. For any message (sequence of one or more bits), let w = M · parity(M ). You should be able to conﬁrm
that parity(w) = 0. Parity lets us...
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- Fall '13