Soft decoding , dual BCH codes , and better list - decodable ε - biased codes ∗

Explicit constructions of binary linear codes that are efficiently list-decodable up to a fraction (1/2−ε) of errors are given. The codes encode k bits into n = poly(k/ε) bits and are constructible and list-decodable in time polynomial in k and 1/ε (in particular, in our results ε need not be constant and can even be polynomially small in n). These results give the best known polynomial dependence of n on k and 1/ε for such codes. Specifically, they are able to achieve n 6 O(k/ε) or, if a linear dependence on k is required, n 6 Õ(k/ε), where γ > 0 is an arbitrary constant. The best previously known constructive bounds in this setting were n 6 O(k/ε) and n 6 O(k/ε). Non-constructively, a random linear encoding of length n = O(k/ε) suffices, but no sub-exponential algorithm is known for list decoding random codes. The construction with a cubic dependence on ε is obtained by concatenating the recent Parvaresh-Vardy (PV) codes with dual BCH codes, and crucially exploits the soft decoding algorithm for PV codes. The result with the linear dependence on k is based on concatenation of the PV code with an inner code of good minimum distance. In addition to being a basic question in coding theory, codes that are list-decodable from a fraction (1/2−ε) of errors for ε→ 0 are important in several complexity theory applications. For example, the construction with near-cubic dependence on ε yields better hardness results for the problem of approximating NP witnesses. In addition, the codes constructed have the property that all nonzero codewords have relative Hamming weights in the range (1/2− ε, 1/2 + ε); this ε-biased property is a fundamental notion in pseudorandomness. ∗A preliminary conference version of this paper was presented at the 23rd Annual IEEE Conference on Computational Complexity (CCC 2008). †Computer Science Department, Carnegie Mellon University, Pittsburgh, PA. Part of this work was done while visiting the School of Mathematics, Institute for Advanced Study, Princeton, NJ. Research supported in part by a Packard fellowship, and NSF grants CCF-0343672, CCF-0953155. [email protected] ‡Department of Computer Science and Engineering, University at Buffalo, State University of New York, Buffalo, NY, 14620. Research supported in part by startup funds from University at Buffalo and by NSF CAREER award CCF-0844796. [email protected]