I'm learning about CRC and Hamming distance and I have three questions. Lets say we have binary code described by ($+$ refers to sum modulo $2$):
\begin{alignat*}{1} a_1 &+ a_2 &+ a_3 &+ a_4 &+ a_5 &= 0\\ a_6 &+ a_7 &+ a_8 &+ a_9 &+ a_{10} &= 0\\ a_1 &+ a_6 &+ a_{11} &&&= 0\\ a_2 &+ a_7 &+ a_{12} &&&= 0\\ a_3 &+ a_8 &+ a_{13} &&&= 0\\ a_4 &+ a_9 &+ a_{14} &&&= 0\\ a_5 &+ a_{10} &+ a_{15} &&&= 0 \end{alignat*} We received: $11011\ 01010\ 10111$
So if I would like to check minimal Hamming distance, then I should compare each codeword with others, right?
$\begin{pmatrix} 1&1&1&1&1& & & & & & & & & & \\ & & & & &1&1&1&1&1& & & & & \\ 1& & & & &1& & & & &1& & & & \\ &1& & & & &1& & & & &1& & & \\ & &1& & & & &1& & & & &1& & \\ & & &1& & & & &1& & & & &1& \\ & & & &1& & & & &1& & & & &1\\ \end{pmatrix}$
I have done so, and minimal Hamming distance $ = 6$. Is there any faster method?
Second question: how many distortions causing undetectable errors? Is there any formula for that?
And last one, how to compute redundancy using above information. I'm using formula:
$ R = \dfrac{n-k}{n}$. In above example $n=15$, but I don't understand how to obtain $k$.