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.

Highlights

  • List decodable codes are extensively studied in Coding Theory and Theory of Computer

  • In the paragraph below we define list-decodable codes, using a functional view, which is more convenient for this paper

  • We say that Enc : {0, 1}k → {0, 1}n, is (p, L)-list decodable, if there exits a function Dec which given y ∈ {0, 1}n, Dec(y) produces a list of size L containing all elements m ∈ {0, 1}k such that δ(y, Enc(m)) ≤ p, (here δ(x, y) is the relative hamming distance of x and y)

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Summary

Introduction

List decodable codes are extensively studied in Coding Theory and Theory of Computer. Binary codes achieving rate approaching 1 − H(p), are known for restricted classes of channels. Natural examples of complexity classes are polynomial size circuits and logarithmic space branching programs Note that these two classes are nonuniform, and it is more natural to use nonuniform classes, as such classes trivially contain channels C where EC is constant (meaning that there is a fixed error vector e such that C(z) = z ⊕ e). 45:3 called “additive channels” and as they are the simplest form of adversarial behavior, it makes sense that we allow them in any class of computationally bounded channels Another advantage of using nonuniform classes of channels, is that it is sufficient to consider deterministic channels, in order to obtain security against randomized channels. This is because by averaging, if there is a computationally bounded randomized channel that is able to prevent decoding on some message m, we can fix its random coins and obtain a deterministic channel (which is hardwired with a good choice of random coins)

Stochastic codes
A Monte-Carlo construction of stochastic codes for poly-size circuits
Our results
Explicit stochastic codes for poly-size circuits
Perspective
Some related work
Overview of the technique
Codes for the setup of shared private randomness
A more efficient reduction for online logspace and AC0
Stochastic codes for bounded channels
Pseudorandom stochastic inner codes
New constructions of pseudorandom weak inner codes
Inner stochastic codes for AC0
Pseudorandom generators
Poly-size circuits
Oblivious read once branching program
Constant depth circuits
Error-Correcting Codes
Codes for balanced errors
List recoverable codes
Averaging Samplers
Almost t-wise permutations
Inner Stochastic codes
PRGs give inner stochastic codes
Inner Stochastic codes for circuits and ROBPs
Inner stochastic codes for AC0 channels
The construction of stochastic codes
Poly-size circuit channels
Online logspace channels
Constant depth channels
Analyzing the construction
Milestones for correct decoding
Milestones Lemma implies Theorem 31
The hiding lemma
Hiding lemma implies milestones lemma
Proof of Lemma 39
Proof of Lemma 40
Proof of Theorem 14
Full Text
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