The Structure of Polarized Channels via Explicit Parameters

The polarization of the BSC(γ 1 ) with a BSC(γ 2 ) is characterized explicitly for γ 1 , γ 2 ∈ [0, 1/2]. The polarization yields a channel W - which is a BSC(λ), and a channel W + which is composed of a BSC(ξ) with probability 1 - λ and a BSC(φ) with probability λ. The parameters λ, ξ and φ are functions of γ 1 and γ 2 . For a general binary-input, output-symmetric, discrete, memoryless (BMS) channel W, a simple method is identified for constructing polar codes based on the fact that each polarized channel is defined by a mutual information profile, and is comprised of sub-channel components, similar to results by [Pedarsani et al., 2011; Tal and Vardy, 2013]. Algebraic polar transforms may be applied recursively to each sub-channel component. As an example, polar codes are constructed for a hybrid BMS channel with an erasure probability ε, a bit-flip probability γ, and capacity C(ε, γ) = (1 - ε)(1 - h b (γ)) where h b (x) -x log 2 (x)-(1-x) log 2 (1-x). Based on the structure of polarization via explicit parameters, relations regarding the information density and channel dispersion V (W) are analyzed for polarized channels, including the super-martingale property of V (W). The analysis depends on second-order terms involving the function ψ(x) x log 2 2 x + (1 - x) log 2 2 (1 - x).