This is a research-expository paper. It deals with
complexity issues in the theory of linear block codes. The main
emphasis is on the theoretical performance limits of the
best known codes. Therefore, the main subject of the paper are
families of asymptotically good codes, i.e., codes whose rate and
relative ... more >>>

We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $\epsilon > 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a *polynomial* in $1/\epsilon$. Polar coding gives the *first known explicit construction* with rigorous ... more >>>