List decoding of noisy Reed-Muller-like codes

Coding theory has played a central role in the development of computer science. One critical point of interaction is decoding error-correcting codes. Firstand 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, the first steps are taken toward extending the quick randomized decoding tools of RM(1) into the realm of quadratic binary and, equivalently, Z4 codes. The 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 , a list is found that is a superset of all Hankel codewords φ with | 〈s, φ〉 | ≥ (1/k) ‖s‖, in time poly(k, log(N)). A new and simple formulation of a known Kerdock code is given as a subcode of the Hankel code which leads to two immediate corollaries. First, the new Hankel list-decoding algorithm covers subcodes, including the new Kerdock construction, so it can list-decode Kerdock, too. Furthermore, because dot products of distinct Kerdock vectors have small magnitude, a quick algorithm is obtained for finding a sparse Kerdock approximation. That is, for k small compared with 1/ √ N and for > 0, in time poly(k log(N)/ ), a k-Kerdock-term approximation e s to s is found with Euclidean error at most the factor (1 + +O(k/ √ N)) times that of the best such approximation.