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).