existence of a linear code with special parameters

Question

Prove that there is an $[nN, (n −1)K, 2D]$-linear code over $F_q$ (a finite field) whenever there is an $ [N, K, D]$-linear code over $F_{q^{n−1}}$ .

All I can think of is putting $n-1$ copies of every codeword after itself...

Answer 1

Hint: Encode each element $z\in\Bbb{F}_{q^{n-1}}$, linearly over $\Bbb{F}_q$, to a vector $\phi(z)\in\Bbb{F}_q^n$ by first representing $z$ w.r.t. a chosen basis (giving you a vector $z\in\Bbb{F}_q^{n-1}$) and then adding a checksum symbol to make sure that each non-zero vector $\phi(z)$ has weight $\ge2$.