Extending extended codes

I picked the following definition of extended codes:

Let C be a [n, k, d] code over Fq. Ĉ = {x1...x_{n+1} in (Fq)^(n+1) | x1...xn in C and x1 + ... + x_{n+1} = 0} is called the extended code of C. (sorry about the poor formatting).

As generator matrix is computed by adding a extra column on the right which contains minus the sum of all elements of the row, it means that if one extends an extended code, it will add an extra column of zeroes...