List decoding of noisy Reed-Muller-like codes

First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are two fundamental error-correcting codes which arise in communication as well as in probabilistically-checkable proofs and learning. In this paper, we take the first steps toward extending the quick randomized decoding tools of RM(1) into the realm of quadratic binary and, equivalently, Z_4 codes. Our main algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and RM(2). That is, given signal s of length N, we find a list that is a superset of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times the norm of s, in time polynomial in k and log(N). We also give a new and simple formulation of a known Kerdock code as a subcode of the Hankel code. As a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm for finding a sparse Kerdock approximation. That is, for k small compared with 1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such approximation

List-decoding reed-muller codes over small fields

We present the first local list-decoding algorithm for the rth order Reed-Muller code RM(r,m) over F2 for r ≥ 2. Given an oracle for a received word R: Fm2 → F2, our random-ized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2−r − ε) from R for any ε> 0 in time poly(mr, ε−r). The list size could be exponential in m at radius 2−r, so our bound is op-timal in the local setting. Since RM(r,m) has relative dis-tance 2−r, our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polyno-mial in the block-length, we show that list-decoding is pos-sible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(21−r), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2,m) codes are list-decodable up to radius η for any constant η < 1 2 in time polynomial in the block-length. Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1