All referee remarks are taken into account:
 
1) Abstract: "the covering radius of an arbitrary completely regular code in the $n$-dimensional rectangular grid is at most $2n$" "arbitrary" should be "irreducible" here.
--corrected

2) I think that the monotonicity should be included in the abstract as one of the main results.
--included. In section 2, the result is presented as Theorem 1.
3) In the introduction, it should be mensioned the same result for distance-regular graphs.
--mentioned

4) End of Sect. 1: define equivalence for c.r.c. (used in the beginning of Sect.2) --defined

5) p.148, Proof of Lemma 3 "By assumption, $c_i=c_{i+1}$. " But the assumption is different: $c_i=c_{i-1}$.          --corrected to $c_i=c_{i+1}$ in the Proof as well as in the Lemma

6)148+9: "Then a) is true" ->  "Therefore a) is true", or "It follows that a) is true", ... --corrected

7) Section 4, first line "can be repeated if the intersection matrix of a completely regular code"  --corrected

8) \bigcup should be \cup everywhere   --corrected

9) "see" -> ``see'', "sees" -> ``sees'' everywhere   --corrected

10) 148-3: " It is left to consider" -> "It remains to consider"   --corrected

11) 150-5 "(10, (11)" ->  "(10), (11)"   --corrected

12) Theorem 3: 4 should be in $4$    --corrected  (now it is Theotem 4)

13) Last two lines before References: \{( -> \{   --corrected

14) Last paragraph before References:
"We put ... "
These two vertices are already in $S_2(x)$; there is no need to put them there.
As an option: "We consider ..."     --corrected

15) Proof of Lemma 6: the overindices should not be in bold.   --corrected