Hello everyone reading this. I seem to have a problem understanding weights in Coding Theory, and will attempt to provide a solution to a problem - please correct me where I am wrong.
Consider the finite field $F_9$, and x a solution to the equation $X^2+1=0$. Let C be the code with parity-check matrix
$h=\begin{bmatrix} 1 & 0 & 1 & x & 1\\ 0 & 1 & 1 & 1 & x \end{bmatrix}$
Find two code words of minimum weight.
My attempt at a solution: First, the code words shouldn't be zero. Any codeword $c=(c_1,c_2,c_3,c_4,c_5)$ will satisfy $ch=0$, that is $$ c_1+c_3+xc_4+c_5=0,c_2+c_3+c_4+xc_5=0 $$ Solving for $c_1,c_2$ gives the basis $$ a=(-1,-1,1,0,0),b=(-x,-1,0,1,0),c=(-1,-x,0,0,1) $$ So the dimension is 3, is this right? Now, the weights. The weight is the number of nonzero elements, so I see that $w(a)=w(b)=w(c)=3$. So are $a,b$ just the required code words? And how does $x$ come into play?
Thanks in advance for any replies...