Extremality properties for Gallager's random coding exponent
Mine Alsan · 2012
We describe certain extremality properties for Gallager's reliability function Eofor binary input symmetric DMCs. In particular, we show that amongst such DMC's 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) - pE'0(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 Arıkan's polar transformations in the sense that their polar transforms W+and W-has the largest and smallest difference in their Eovalues.