Mine Alsan

Extremal Channels of Gallager's

We study the extremality of the binary erasure channel and the binary symmetric channel for Gallagers reliability function E0 of binary input discrete memoryless channels evaluated under the uniform input distribution from the aspect of channel polarization. In particular, we show that amongst all binary discrete memoryless channels of a given E0(ρ) value, for a fixed ρ ≥ 0, the binary erasure channel and the binary symmetric channel are extremal in the evolution of E0 under the one-step polarization transformations.

E_{0}

We give a simple proof of Arikans polarization phenomenon that uses only elementary methods. Using the same method, we show that Arikans construction also polarizes non-stationary memoryless channels in the same way it polarizes stationary memoryless channels. This is a new result.

Under the Basic Polarization Transformations

We describe certain extremality properties for Gallagers reliability function E_o for binary input symmetric DMCs. In particular, we show that amongst such DMCs whose E₀(ρ₁) has a given value for a given ρ₁, the BEC and BSC have the largest and smallest value of the derivative of Eo(ρ₂) for any ρ₂ ≥ ρ₁. As the random coding exponent is obtained by tracing the map ρ → (E₀(ρ), E₀(ρ) - pE₀(ρ)) this conclusion includes as a special case the results of [1]. Furthermore, we show that amongst channels W with a given value of E₀(ρ) for a given ρ the BEC and BSC are the most and least polarizing under Arıkans polar transformations in the sense that their polar transforms W⁺ and W^- has the largest and smallest difference in their E_o values.

A simple proof of polarization and polarization for non-stationary channels

We study partial orders on the information sets of polar codes designed for binary discrete memoryless channels. We show that the polar transform defined by Arikan preserves `symmetric convex/concave orders. While for symmetric channels this ordering turns out to be equivalent to the stochastic degradation ordering already known to order the information sets of polar codes, we show that a strictly weaker partial order is obtained when at least one of the channels is asymmetric. We also discuss two tools which can be useful for verifying this ordering: a criterion known as the cut criterion and channel symmetrization.

Extremality properties for Gallager's random coding exponent

We give a simple proof of Arıkans polarization phenomenon that uses only elementary methods. Using the same method, we show that Arıkans construction also polarizes non-stationary memoryless channels in the same way it polarizes the stationary memoryless channels.

A novel partial order for the information sets of polar codes over B-DMCs

We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Arikans polar transform. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.