I'm currently working my way through "Fundemental of Error Correcting Codes" by Huffman and Pless. However I'm stuck on exercise 95 which states:
Suppose we have a linear code $\mathcal{C}$ with $B_q(n, d)$ codewords ($B_q(n, d)$ being the maximum number of codewords of a linear code with length $n$ and minimum distance at least $d$), then the exercise is to show that $\mathcal{C}$, has covering radius $d-1$ or less, meaning: $$ \mathbb{F}_q^n = \bigcup_{c \in \mathcal{C}} B_{d - 1}(c) $$ where $B_{r}(y) = \{x \in \mathbb{F}_q^n | d(x, y) \leq r\}$.
I have included their trick for when $\mathcal{C}$ is non-linear below:
Suppose $x \in \mathbb{F}_q^n \setminus \bigcup_{c \in \mathcal{C}} B_{d-1}(c)$, where $B_{d-1}$ are hamming balls of radius $d-1$. Then one can add $x$ to $\mathcal{C}$, (which contradicts the fact that $\mathcal{C}$ had the maximum number of codewords) but this doesn't exactly work in the linear case, since $\mathcal{C}' = \mathcal{C} \cup \{x\}$ is not necessarily a linear code. My next idea is to let $\mathcal{C}' = \mathcal{C} + span\{x\}$, but then i can't figure out the minimum distance of $\mathcal{C}'$. Any help would be much appricated.
Edit:
Sorry for the inconvenience, I have updatede the question.
More or less the same argument still works. This is a typical use of linearity in coding theory.
Assume, as in your summary of the non-linear argument, that there exists a vector $x\in\Bbb{F}_q^n$ that does not belong to any ball of radius $d-1$ centered at a codeword. As you guessed, it follows that the linear code $$ \mathcal{C}^+=\{y+ax\mid y\in\mathcal{C}, a\in\Bbb{F}_q\} $$ will still have minimum Hamming distance $\ge d$, in violation of maximality of $\mathcal{C}$.
We see this as follows. Recall that for linear codes the minimum distance is equal to the minimum (non-zero) weight. So a contrapositive assumption is that there exists a word of the prescribed form $$ z=y+ax\in\mathcal{C}^+ $$ of Hamming weight at most $d-1$. As the minimum weight of vectors of $\mathcal{C}$ is $d$, it follows that $z\notin\mathcal{C}$. In other words, $a\neq0$. But the Hamming weight does not change, when a codeword is multiplied by a non-zero scalar. It follows that the weight of the word $$ -a^{-1}z=(-a^{-1}y)-x\in\mathcal{C}^+ $$ is also $\le d-1$. But then $$ d_H(-a^{-1}y,x)=w(-a^{-1}y-x)\le d-1. $$ This is a contradiction because it implies that the vector $x$ is in the ball $B_{d-1}(-a^{-1}y)$ of radius $d-1$ centered at the codeword $-a^{-1}y\in\mathcal{C}$.
Hence we can conclude that the covering radius of a maximal linear code of minimum distance $d$ is strictly less than $d$.
I was a bit disappointed that I could not find it here already, as the technique is pretty standard. I do recall having to think it through as a student of coding theory, but this becomes second nature quickly enough! Here Kodlu is using this result in a different context (+1).