Binary code and Hamming distance 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 &amp;+ a_2 &amp;+ a_3 &amp;+ a_4 &amp;+ a_5 &amp;= 0\\
a_6 &amp;+ a_7 &amp;+ a_8 &amp;+ a_9 &amp;+ a_{10} &amp;= 0\\
a_1 &amp;+ a_6 &amp;+ a_{11} &amp;&amp;&amp;= 0\\
a_2 &amp;+ a_7 &amp;+ a_{12} &amp;&amp;&amp;= 0\\
a_3 &amp;+ a_8 &amp;+ a_{13} &amp;&amp;&amp;= 0\\
a_4 &amp;+ a_9 &amp;+ a_{14} &amp;&amp;&amp;= 0\\
a_5 &amp;+ a_{10} &amp;+ a_{15} &amp;&amp;&amp;= 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?

$\begin{pmatrix}
1&amp;1&amp;1&amp;1&amp;1&amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\
 &amp; &amp; &amp; &amp; &amp;1&amp;1&amp;1&amp;1&amp;1&amp; &amp; &amp; &amp; &amp; \\
1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; \\
 &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; \\ 
 &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; \\
 &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; \\
 &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1&amp; &amp; &amp; &amp; &amp;1\\
\end{pmatrix}$

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 $k$.
In a linear code, the minimum distance is the same as the minimum weight of a non-zero codeword. It's a nice exercise to prove this, and reduces the computation time quadratically.

Computing the minimum distance of a code given the generator matrix (or the parity check matrix) is NP-hard, that is, given a generator matrix (or a parity check matrix) and an integer $w$, it is NP-complete to determine whether the minimum distance of the code generated by the matrix is at most $w$. The task could become easier if the code has some structure, but for arbitrary codes it is quite difficult.