Linear code over $F_q$ exists always

Question

Why is there always an $[n, n-1, 2]$ linear code over $F_q$ for any $n \geq 2$?

Answer 1

Well, the code is equivalent to a code with systematic generator matrix $$G = \left(\begin{array}{llllll} g_{11} & g_{12}& \ldots & g_{1,n-2} & g_{1,n-1} & g_{1n}\\ 0 & g_{22}&\ldots & g_{2,n-2} & g_{2,n-1} & g_{2n}\\ \vdots \\ 0 & 0 & \ldots & 0 & g_{n-1,n-1} & g_{n-1,n} \end{array}\right),$$ where the (diagonal) entries $g_{11},\ldots,g_{n-1,n-1}$ are nonzero. From here you see directly that the code has dimension $n-1$ and minimum Hamming distance 2.

Answer 2

This code satisfies the Singleton bound, so it is MDS. So its dual code is also MDS. Consider the parameters of its dual code. Do you recognise it?