Effective version of the Shannon theorem: Polar codes

A sequence of codes Cn = fn(Mn), n ≥ 1 is said to attain the transmission rate I(W ) on the channel W if for any ε > 0 there exists a sufficiently large n0 such that for all n ≥ n0 bothR > I(W )−ε and Pe(Cn) ≤ ε. Below we construct a sequence of linear binary codes Cn, n ≥ 1 of length N = 2 that attains the rate I(W ) of the channelW . For symmetric binary-input channels these codes are capacity-achieving.