The analogue of the Varshamov inequality need not hold for instance for $\lvert\Sigma\rvert=q=6$: For $n=4$ and $d=3$ it would give a code $C\subseteq\Sigma^4$ with $d(C)\ge3$ and $\lvert C\rvert=36$. But such a code gives a pair of orthogonal Latin squares of order $6$, which is known not to exist (by Tarry's negative answer to Euler's thirty-six officers problem).

The analogue of the Varshamov inequality need not hold for instance for $\lvert\Sigma\rvert=q=6$: For $n=4$ and $d=3$ it would give a code $C\subseteq\Sigma^4$ with $d(C)\ge3$ and $\lvert C\rvert=36$. But such a code gives a pair of orthogonal Latin squares of order $6$, which is known not to exist (by Tarry's negative answer to Euler's thirty-six officers problem).

(The construction of the orthogonal pair $A,B$ is as follows: Since $d(C)\ge3$, the map $C\to\Sigma^2$, $(i,j,a,b)\mapsto(i,j)$ is bijective. For $(i,j,a,b)\in C$ let $A$ and $B$ have entry $a$ and $b$ in the $i$-th row and $j$-th column, respectively.)