Local List-Decoding of Reed-Muller Codes, Study notes of Number Theory

Download Local List-Decoding of Reed-Muller Codes and more Study notes Number Theory in PDF only on Docsity! CS225: Pseudorandomness Prof. Salil Vadhan Lecture 21: Local List-Decoding April 26, 2007 Based on scribe notes by Kevin Matulef. 1 Local List Decoding In previous lectures, we talked about a local decoding algorithm as a probabilistic algorithm which, when given oracle access to a function g close to some codeword f̂ , and given an input x, would output f̂(x) with high probability. Pictorially, this is shown below: Dec f(x)x g oracle access In order to decode from distances close to 1/2 with a binary code, we would like to formulate a notion of local list-decoding. This is slightly trickier to define, since for any function g, there may be several codewords f̂1, f̂2, ..., f̂s that are close to g. So what should our decoding algorithm do? One option would be for the decoding algorithm, on input x, to output a list f̂1(x), ˆf2(x), ...f̂s(x). However, rather than outputing each of these values, we want to be able to specify to our decoder which f̂i(x) to output. We do this with a two-phase decoding algorithm. The probabilistic algorithms that accomplish these phases will be referred to as Dec1 and Dec2: 1. Dec1, using g as an oracle, returns a list of advice strings a1, a2, ..., a2, which can be thought of as “labels” for each of the codewords close to g. 2. Dec2 (again, using oracle access to g), takes input x and ai, and outputs f̂i(x). The picture for Dec2 is much like our old decoder, but it takes an extra input ai corresponding to one of the outputs of Dec1: g oracle access f_i(x) a_i x Dec_2 1 More formally, Definition 1 A local δ list-decoding algorithm for a code Enc is a pair of probabilistic oracle algorithms (Dec1,Dec2) such that for all received words g and all codewords f̂ = Enc(f) with ∆(f̂ , g) < δ, the following holds. With probability greater at least 1/2 over (a1, ..., as) ← Decg1, there exists an i ∈ [s] such that ∀x,Pr[Decg2(x, ai) = f(x)] ≥ 2/3. To help clarify this definition, we make the following remarks. First, we don’t require that for all j,Decg2(x, aj) are codewords, or even that they’re close to s; in other words some of the aj ’s may be junk. Second, we don’t explicitly require a bound on list size s, but certainly it is less than time(Dec1). As we did for locally (unique-)decodable codes, we can define a local δ list-decoding algorithm for codeword symbols, where Dec2 should recover arbitrary symbols of the codeword f̂ rather than the message f . As before, this implies the above definition if the code is systematic. Two lectures ago, we explained how having a local decoding algorithm and a worst-case hard function implied having an average-case hard function. Similarly, if we have a local list-decoding algorithm, we can make the following statement: Proposition 2 If Enc has a local δ-list decoding algorithm (Dec1,Dec2), and f is worst-case hard for non-uniform time t = t(`), then f̂ = Enc(f) is (t′, δ)-hard, where t′ = t/time(Dec2). Proof: Suppose that f̂ is not (t′, δ)-hard. Then some algorithm A running in time t′ computes f̂ with error probability smaller than δ. But if Enc has a local δ list-decoding algorithm, then (with A playing the role of g) that means there exists ai (one of the possible outputs of Dec A 1 ), such that DecA2 (·, ai) computes f(·) everywhere. The running time of DecA2 (·, ai) is time(A) · time(Dec2) = t. Note that here we are using nonuniformity crucially to hardwire ai as advice, in order to select the right function from the list of possible decodings. 2 Local List-Decoding Reed–Muller Codes Theorem 3 There is a universal constant c such that the m-variate Reed–Muller code of degree d over a finite field F can be locally (1− ε)-list decoded in time poly(|F|,m) for ε = c √ d/|F|. Note that the distance at which list-decoding can be done approaches 1 as |F|/d→∞. It matches the bound for Reed–Solomon codes (up to the constant c) with the benefit of sublinear-time de- coding for large enough m; however, the rate is worse than for Reed–Solomon codes. Proof: Suppose we are given an oracle g : Fm → F that is (1−ε) close to some unkown polynomial p : Fm → F, and that we are given an x ∈ Fm. Our goal is is describe two algorithms, Dec1 and Dec2, where Dec2 is able to compute p(x) using a piece of Dec1’s output (i.e. advice). 2