Equal-image-size source partitioning: Creating strong Fano's inequalities for multi-terminal discrete memoryless channels

This paper introduces equal-image-size source partitioning, a new tool for analyzing channel and joint source-channel coding in a multi-terminal discrete memoryless channel environment. Equal-image-size source partitioning divides the source (combination of messages and codewords) into a sub-exponential number of subsets. Over each of these subsets, the exponential orders of the minimum image sizes of most messages are roughly equal to the same entropy term. This property gives us the strength of minimum image sizes and the flexibility of entropy terms. Using the method of equal-image-size source partitioning, we prove separate necessary conditions for the existence of average-error and maximum-error codes. These necessary conditions are much stronger than the standard Fano's inequality, and can be weakened to render versions of Fano's inequality that apply to codes with non-vanishing error probabilities. To demonstrate the power of this new tool, we employ the stronger average-error version of Fano's inequality to prove the strong converse for the discrete memoryless wiretap channel with decaying leakage, which heretofore has been an open problem.