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My question is about the following result about list-decoding of

Sudan, Trevisan and Vadhan.

enter image description here (The formulation is taken from Shuichi Hirahara's paper.)

I do not understand how this is possible. I think this result contradicts to a simple information reasoning:

  1. Assume $\epsilon = \frac{1}{10}$ for example. Then the algorithm works in time $\text{poly} (\log N)$ and the cardinality of list $L$ is $\text{poly} (\log N)$ too.
  2. Fix some x and random bits. We say that the algorithm works correctly in this case if it works correctly (the output list contains $x$) for $r=Enc(x)$.
  3. For some $N$ we can fix random bits of the decoding algorithm such that this algorithm is transformed to a deterministic circuit $D$ that works correctly for most inputs. Denote by $A$ the set of ``good'' inputs; the cardinality of $A$ is greater than $2^{N-1}$.
  4. This circuit $D$ allows by asking $\text{poly} (\log N)$ Yes/No questions to determine a string $x \in A$ : first, the circuit asks $\text{poly} (\log N)$ such questions about bits of $Enc(x)$; then circuit outputs $\text{poly} (\log N)$-list containing $x$ - now we need just $O(\log \log N)$-bits to determine $x$.
  5. However, the yes/no-question algorithm above can determine only $2^{\text{poly} (\log N)}$ strings that is much less then cardinality of $A$.

Why am I wrong?

UPD: I have understood. Circuits $C_j$ has access to $r$ so it is impossible to determine $x$ by my way. Sorry for the confusion.

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  • $\begingroup$ When you say "works for most inputs" and "good inputs", what do you mean by "input"? Do you refer to the string $r$ or to the strings $x$ that are close to $r$? If you refer to the strings $x$, then there are not so many such strings: for a fixed $r$, there are only $O(1)$ such $x$'s (for your choice of $\epsilon$). If you refer to $r$, then I am not sure how you can determine it using a small number of yes/no questions. $\endgroup$
    – Or Meir
    Commented Oct 20, 2022 at 21:29
  • $\begingroup$ @OrMeir Inupts are strings $x$. The algorithm works correctly on input $x$ if it works for $r=Enc(x)$. $\endgroup$ Commented Oct 21, 2022 at 3:05

1 Answer 1

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I don't think Step 4 works. The circuit $C_j^{\mathrm{Enc}(x)}$ takes $i \in [N]$ as input and outputs the $i$-th bit of $x$. For each $i \in [N]$, the circuit asks $\mathrm{poly}(\log N)$ queries to bits of $\mathrm{Enc}(x)$, but the queries depend on $i$. In order to get a short list that contains $x$, you need to answer all the queries for every $i \in [N]$, which can be as many as $O(N \cdot \mathrm{poly}(\log N))$ in total (and cannot be bounded by $\mathrm{poly}(\log N)$).

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  • $\begingroup$ The circuit that I defined is not any $C_j$ from the theorem. It is a derandomization (not full, only for $x \in A$) of Dec that not depend on $i$ $\endgroup$ Commented Oct 21, 2022 at 8:21

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