  An $n$-length binary word is $q$-decreasing, $q\geq 1$, if every of its
length maximal factor of the form $0^a1^b$ satisfies $a=0$ or $q\cdot a > b$.We
show constructively that these words are in bijection with binary words having
no occurrences of $1^{q+1}$, and thus they are enumerated by the
$(q+1)$-generalized Fibonacci numbers. We give some enumerative results and
reveal similarities between $q$-decreasing words and binary words having no
occurrences of $1^{q+1}$ in terms of frequency of $1$ bit. In the second part
of our paper, we provide an efficient exhaustive generating algorithm for
$q$-decreasing words in lexicographic order, for any $q\geq 1$, show the
existence of 3-Gray codes and explain how a generating algorithm for these Gray
codes can be obtained. Moreover, we give the construction of a more restrictive
1-Gray code for $1$-decreasing words, which in particular settles a conjecture
stated recently in the context of interconnection networks by E\u{g}ecio\u{g}lu
and Ir\v{s}i\v{c}.
