Suppose exist a $[n , k , d ]$ liner block code like $C$ can you get me hints to prove exist a $ [ n , k , d-1 ] $ liner block code like $C' $?

Question

Suppose exist a $[n , k , d ]$ liner block code like $C$ from $\Bbb{F_q}^n $ can you get me hints to prove exist a $ [ n , k , d-1 ] $ liner block code like $C' $ from $\Bbb{F_q}^n $?

Answer 1

We know that $$d = \min_{c \in C/\{0\}}\|c\|$$

Let $c \in C$ and $d = \|c\|$.

We can see that $c \ne 0$, and we find $c_2,\ldots, c_k$ : $(c, c_2,\ldots, c_k)$ - basis in $C$.

Let $c_i \ne 0 $ be some coordinate of $c$.

Let $c' = c - c_i*e_i$, $e_i$ is $i$ unity vector.

We can see that $c' \notin C$.

If $d > 1$ then $C'$ with basis $(c', c_2,\ldots,c_k)$ is $[n,k, d - 1]$ linear block code.

If $d = 1$ then $[n, k, d - 1]$ linear block code must be trivial - $C' = \{0\}$