Ju l 2 00 6 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. 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, Z4 codes. Our main algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin and Kushilevitz-Mansour algorithms [GL89, KM91] to the Hankel code [CGL + 05], 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 ϕ with | s, ϕ | 2 ≥ (1/k) s 2 , in time poly(k, log(N)). We then turn our attention to the widely-studied Kerdock codes. We give a new and simple formulation of a known Kerdock code as a subcode of the Hankel code. We then get two immediate corollaries. First, our new Hankel list-decoding algorithm covers subcodes, including the new Kerdock construction, so we can list-decode Kerdock, too. Furthermore, exploiting the fact that dot products of distinct Kerdock vectors have small magnitude, we get a quick algorithm for finding a sparse Kerdock approximation. That is, for k small compared with 1/ √ N and for ǫ > 0, we find, in time poly(k log(N)/ǫ), a k-Kerdock-term approximation s to s with Euclidean error at most the factor (1 + ǫ + O(k 2 / √ N)) times that of the best such approximation.