Let W be a binary-input memoryless symmetric (BMS) channel with Shannon capacity I(W) and fix any \alpha > 0. We construct, for any sufficiently small \delta > 0, binary linear codes of block length O(1/\delta^{2+\alpha}) and rate I(W)-\delta that enable reliable communication on W with quasi-linear time encoding and decoding. ... more >>>

We say a subset C \subseteq \{1,2,\dots,k\}^n is a k-hash code (also called k-separated) if for every subset of k codewords from C, there exists a coordinate where all these codewords have distinct values. Understanding the largest possible rate (in bits), defined as (\log_2 |C|)/n, of a k-hash code is ... more >>>