Let $q$ be an integer and $F_q$ the set $\{0, 1, \ldots , q-1\}$ and let $F_q^n$ be the $q$-ary hypercube of dimension $n$, i.e. the set of vectors whose the $n$ components belong to $F_q$. For $x=(x_1, \ldots ,x_n) \in F_q^n$ and ${y=(y_1, \ldots , y_n) \in F_q^n}$, the {\it Lee distance} between $x$ and $y$ is equal to ${\sum _{i=1}^{n}\min(|x_i-y_i|,q-|x_i-y_i|)}$. Let $C\subseteq F_q^n$; $C$ is called a {\it code}. Given an integer radius $r\geqslant 1$, we consider three types of codes with respect to the Lee distance: an $r$-{\it dominating code} $C$ (also called an $r$-{\it covering code}) is such that any element $x\in F_q^n$ is within distance $r$ from at least one codeword $c\in C$ (then $c$ $r$-{\it dominates}~$x$); an $r$-{\it locating-dominating code} $C$ is (i)~$r$-dominating and (ii)~such that any two elements $x$ and $y$ belonging to $F_q^n \setminus C$ are $r$-dominated by distinct sets of codewords; an $r$-{\it identifying code} $C$ is (i)~$r$-dominating and (ii)~such that any two elements $x$ and $y$ belonging to $F_q^n$ are $r$-dominated by distinct sets of codewords. We look for minimum such codes. We improve the cardinalities of $r$-dominating codes, $r$-locating-dominating codes or $r$-identifying codes of $F_q^n$ for some values of the alphabet size $q\in \{4,5,6\}$, of the dimension $n$ up to~$7$, and the radius $r$ up to~$5$.