Introduction to Cryptography with Coding Theory, 3rd Edition

24.12 Exercises

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

2. An ISBN number is incorrectly written as 0-13-116093-8. 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.

3. The following is a parity check matrix for a binary $[n\text{,}k]$ code $C$:

   $\left(\begin{array}{ccccc}1& 1& 1& 0& 0\\ 1& 0& 0& 1& 0\\ 0& 1& 0& 0& 1\end{array}\right)\text{.}$

   1. Find $n$ and $k$.

   2. Find the generator matrix for $C$.

   3. List the codewords in $C$.

   4. What is the code rate for $C$?

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