At first I wanted to say Hello, because its my first post in this stack exchange!
I've just started my course in coding and cryptography and don't know if my approach is correct after reading the theory. The task was to check if a linear code (22, 13, 11) exists.
Below are my approaches. Would be very thankful for checking if my thinking is correct:
Simple check - minimum distance of linear code $ d \leq n-k+1 $. As I thought it's to trivial I checked next with the Hamming Bound
Hamming Bound.
First I've calculated $ t = 5 $ and $ C $ from the formula $ C \leq \frac{2^n}{\binom{n}{0} + \binom{n}{1} + ...+\binom{n}{t} } $ and the result was $ C \leq 119$ but the code is linear so $ C \leq 118 $. But $k =13 $ so such code does not exist ?
- Last check was with Gilbert–Varshamov bound stating that a linear code exists iff $ \binom{n-1}{0}+...+\binom{n-1}{d-2} \leq 2^{n-k} $ which showed it is not a valid code.
Are those approaches correct? Are there any other methods to check if a linear code exists?