## Abstract

A stochastic code is a pair of encoding and decoding
procedures (Enc, Dec) where \({{\rm Enc} : \{0, 1\}^{k} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{n}}\). The
code is (*p, L*)-list decodable against a class \(\mathcal{C}\) of “channel functions”
\(C : \{0,1\}^{n} \rightarrow \{0,1\}^{n}\) if for every message \(m \in \{0,1\}^{k}\) and every channel
\(C \in \mathcal{C}\) that induces at most *pn* errors, applying Dec on the “received
word” *C*(Enc(*m,S*)) produces a list of at most *L* messages that contain
*m* with high probability over the choice of uniform \(S \leftarrow \{0, 1\}^{d}\). Note
that both the channel *C* and the decoding algorithm Dec do not receive
the random variable *S*, when attempting to decode. The rate of a code
is \(R = k/n\), and a code is explicit if Enc, Dec run in time poly(*n*).

Guruswami and Smith (Journal of the ACM, 2016) showed that for
every constants \(0 < p < \frac{1}{2}, \epsilon > 0\) and \(c > 1\) there exist a constant
*L* and a Monte Carlo explicit constructions of stochastic codes with
rate \(R \geq 1-H(p) - \epsilon\) that are (*p, L*)-list decodable for size \(n^c\) channels.
Here, Monte Carlo means that the encoding and decoding need to share
a public uniformly chosen \({\rm poly}(n^c)\) bit string *Y*, and the constructed
stochastic code is (*p, L*)-list decodable with high probability over the
choice of *Y*.

Guruswami and Smith pose an open problem to give fully explicit (that is not Monte Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper, we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97).

Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against \(O(\log n)\)-space online channels. (These are channels that have space \(O(\log n)\) and are allowed to read the input codeword in one pass.) We also resolve this open problem.

Finally, we consider a tighter notion of explicitness, in which the running
time of encoding and list-decoding algorithms does not increase, when
increasing the complexity of the channel. We give explicit constructions
(with rate approaching \(1 - H(p)\) for every \(p \leq p_{0}\) for some \(p_{0} >0\) ) for
channels that are circuits of size \(2^{n^{\Omega(1/d)}}\) and depth *d*. Here, the running
time of encoding and decoding is a polynomial that does not depend on
the depth of the circuit.

Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the reductions in the proof more efficient, so that we can handle weak classes of channels.

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## Acknowledgements

We are grateful to Swastik Kopparty for pointing us to the Algebraic Geometric codes of Garcia and Stichtenoth, and in particular for pointing us to their description in Shpilka (2009). We thank Noga Ron-Zewi for help with list-recoverable codes. We also thank the anonymous referees for helpful feedback.

Ronen Shaltiel was supported by ERC starting grant 279559, ISF grant 864/11, ISF grant 1628/17 and BSF grant 2010120. Jad Silbak was supported by ERC starting grant 279559 and ISF grant 1628/17.

We are grateful to anonymous referees for helpful comments and corrections.

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Shaltiel, R., Silbak, J. Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels .
*comput. complex.* **30, **3 (2021). https://doi.org/10.1007/s00037-020-00203-w

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### Keywords

- error correcting codes
- computationally bounded channels
- pseudorandomness

### Subject classification

- 68Q01