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. ...
Get Introduction to Cryptography with Coding Theory, 3rd Edition now with the O’Reilly learning platform.
O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.