Dear editor and referees,

We took into account your comments.
We commented some of your remarks below.

Sincerely yours,
P. Ochem and A. Pinlou

--------------------------------------------------
The authors took into account the observations in the first report,
 and I now recommend to accept the revised paper, with minor corrections
 suggested below.

 p. 3, l. 3. "a non-empty words" -> "a non-empty word".
DONE

 p. 4, l. 15. "Theorem 2.(b) for doubled patterns and $k \geq 4$"
 -> "Theorem 2.(b) for doubled patterns with $v(p) \geq 4$"
 (there is no $k$ in Theorem 2 any more).
DONE

 p. 6, l. -6. "correspond to $A_1$" -> "correspond to $y(A_1)$"
 then the same with $A_2$.
DONE

 p. 7, l. -10. Lemma 7 is said to be taken from [7]. I checked that
 reference (which I did not do for the first report), and it seems
 to come from Lemma 6 in that paper, but the statement is slightly
 different. The formula that can be deduced from [7] is
 $C_{t,r,d} \leq C_{t+r(d-1),d}$. Lemma 7 here is a bit stronger, but
 is easy to prove anyway.
DONE : We eventually gave a proof of $C_{t,r,d} \leq C_{t+d,d}$.

 p. 7, l. -6. As I noted in the first report, the reference to the
 radius of convergence $\rho$ in the statement of Lemma 8 is a bit
 confusing, as this radius is obviously equal to $1$. In the more
 general statement of [7] this was not the case. I suggest moving
 the remark (which is made right after the lemma) that the radius
 of convergence of $\Phi_d$ is $1$ before the lemma, and stating the
 lemma with $1$ instead of $\rho$.
DONE

 p. 8, l. 5. You have removed the remark that $\gamma_d$ is decreasing,
 but you still implicitly use it. To avoid this, it is sufficient to
 observe that if $q(k) \geq d$, then
 $|\cal D| \leq C_{t,|w_t|,q(k)} \leq C_{t,|w_t|,d} \leq C_{t+d-1,d}$
 (instead of what is currently right before and right after Lemma 7).
 Then apply it for $d=24$ and $d=48$.
DONE

 p. 8, l. 8. "Therefore $|\cal X| \leq ..." is not completely correct
 as you are counting words up to length $t/2$, not of that exact length.
 You should add $1$ to the first exponent, and $2$ to the second one.
DONE : We have rather abandoned the padding 0's and modified accordingly
all the rest (algorithm, example, decoding, and analysis).

 p. 8, l. -3. "l" -> "\ell" (two occurrences).
DONE

 p. 8, l. -2. "$\bar{g_k}(\ell)$ is decreasing": why ?
DONE we show that the derivative is negative

 p. 8, l. -1. "we can see", I assume that the intended argument is that
 $g_k(q(k))$ is a decreasing function of $k$, but again this does not
 seem obvious. "Seeing" it by computing numerically the first values
 is not a proof!
DONE we show that the derivative is negative

 p. 9, l. 2. "$\ell \geq 48$" -> "$\ell \geq q(k)$".
DONE

 p. 9, l. 12 to 30. The set named $\cal A_{|p|}$ clearly depends on the
 $a_i$, that is on $p$ itself, not just its length. Therefore it should
 be named $\cal A_p$ instead (5 occurrences).
DONE

 p. 9, l. 13. "of a pattern of length $|p|$" -> "of $p$".
DONE

 p. 9, l. -5. Delete ", $k=4$ and $24 \leq |p| \leq 99$", as $k$ and $p$
 are not used in this statement.
DONE

 p. 9, l. -4. "Bound $g_k(\ell)$" -> "Bound on $g_k(\ell)$".
DONE

 p. 10, l. 6 and 7. As a factor $2$ is needed in the bound for $\cal X$
 (see above), it should also appear in both inequalities here.
DONE : This is no longer needed as the proof of the initial bound
in the analisys of $\cal X$ has been patched.

------------------------------------------------------

The authors appear to have taken into account all of the reviewers'
observations. The issues in the proof, in particular around Section
4.3.3, have all been resolved. Once again, I recommend the paper for
publication.
The following minor corrections and suggestions may further improve the
quality of the paper, which does not need further revision anyway.

page 3, l.1: a word w -> a word w over the alphabet Sigma
DONE

page 5, l.-15: of each step -> at each step
DONE

page 9, l.1 and thereafter: the new notation for occurrences (with a
morphism hi) conflicts with the definition of the map hk here. This may
cause confusion; I advise to change either notation for the sake of
readability.
DONE : renamed h_k into s_k

page 9, l.-3: It seems that the definition of the bj coefficients, and
therefore of what the authors call A|p|, actually depend on p, or at
least on its Parikh vector (a1,a2,a3,a4), rather than just on its length
|p|. This does not affect the correctness of the proof, but the notation
should show this dependence (e.g., A(a1,a2,a3,a4) instead of A|p|), and
some sentences in the subsequent paragraphs should be modified accordingly.
DONE

page 10, line -8: Two consecutive sections both having name "Conclusion"
is a bit odd, so I would name 4.4 "End of the proof" or something similar.
DONE