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Publication# Extremality Properties for Gallager's Random Coding Exponent

Abstract

We describe certain extremality properties for Gallager's reliability function E-0 for binary input symmetric DMCs. In particular, we show that amongst such DMC's whose E-0(rho(1)) has a given value for a given rho(1), the BEC and BSC have the largest and smallest value of the derivative of E-0(rho(2)) for any rho(2) >= rho(1). As the random coding exponent is obtained by tracing the map rho -> (E-0'(rho), E-0(rho) - rho E-0'(rho)) this conclusion includes as a special case the results of [1]. Furthermore, we show that amongst channels W with a given value of E-0(rho) for a given rho the BEC and BSC are the most and least polarizing under Arikan's polar transformations in the sense that their polar transforms W+ and W- has the largest and smallest difference in their E-0 values.

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Related concepts (1)

Reliability engineering

Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specified period of time. Reliability is closely related to availability, which is typically described as the ability of a component or system to function at a specified moment or interval of time. The reliability function is theoretically defined as the probability of success at time t, which is denoted R(t).