Coding Theory and Generating a matrix My question is: How can I construct a generator matrix for a one-error correcting linear code with 8 codewords? What would the values of the parameters [n,k] be and what would the size of the parity check matrix be?
Does this mean that I need to find a Code that is one error-correcting? 
I think this means that the minimum distance between two of the vectors is 3?
 A: If a $[n,k]$ linear binary code has $8$ codewords, you are being told that
$k = \log_2 8 = 3$; an $[n,k]$ binary code has $2^k$ codewords.  As discussed
in Jyrki Lahtonen's answer to 
another question 
that you asked today, the Hamming spheres of radius $1$ centered 
at the $8$ codewords must not have any vectors in
common.  Since there are $1 + \binom{n}{1} = 1+n$ vectors in each such sphere,
and $2^n$ vectors total, $n$ must be large enough so that
$$2^n \geq 8(n+1) \Rightarrow n \geq 6.$$ 
Note also that you are asked for the generator matrix of the code
which is a $k\times n$ matrix whose row-space (set of all $2^k$ linear
combinations of the rows) is the set of codewords.  So you could take
the vectors suggested to you by Gerry Myerson as the rows of the generator
matrix.
Be aware that shorter codes ($n$ smaller that $9$) can also be found.
If you know about single-error-correcting Hamming codes,
you can write down a $[7,4]$ code or a $[7,3]$ consisting
of all the even weight codewords of the former, that can correct
single errors.
A: Yes, it means you need to find some $n$ and some 8 $n$-bit strings with minimum distance 3, and moreover those 8 strings must form a vector space. 
One really simple way to do this is to use the vector space generated by the strings 111000000, 000111000, and 000000111. Here, $n=9$. But you can probably find something with a smaller value of $n$, and, in this business, smaller is better. 
As for the parameters $n$ and $k$, and the size of the parity check matrix, and the construction of the generator matrix: do you not have a text or some notes defining these things and giving you examples? 
By the way, if this is a homework problem, you are encouraged to use the homework tag.  
A: Since this question is now quite old, I'm going to add the $n=6$ code alluded to:
000000
010101
001111
011010
100110
101001
110011
111100
