24.12 Exercises

Two codewords were sent using the Hamming $[7\text{,}\text{}4]$ code and were received as 0100111 and 0101010. Each one contains at most one error. Correct the errors. Also, determine the 4bit messages that were multiplied by the matrix $G$ to obtain the codewords.

An ISBN number is incorrectly written as 0131160938. Show that this is not a correct ISBN number. Find two different valid ISBN numbers such that an error in one digit would give this number. This shows that ISBN cannot correct errors.

The following is a parity check matrix for a binary $[n\text{,}\text{}k]$ code $C$:
$\left(\begin{array}{ccccc}1& 1& 1& 0& 0\\ 1& 0& 0& 1& 0\\ 0& 1& 0& 0& 1\end{array}\right)\text{.}$
Find $n$ and $k$.
Find the generator matrix for $C$.
List the codewords in $C$.
What is the code rate for $C$?

Let $C=\{(0\text{,}\text{}0\text{,}\text{}0)\text{,}\text{}(1\text{,}\text{}1\text{,}\text{}1)\}$ be a binary repetition code. ...
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