Reliable Communications with Asymmetric Codebooks: An Information Theoretic Analysis of Robust Signal Hashing

In this paper, a generalization of the traditional point-to-point to communication setup, which is named as "reliable communications with asymmetric codebooks", is proposed. Under the assumption of independent identically distributed (i.i.d) encoder codewords, it is proven that the operational capacity of the system is equal to the information capacity of the system, which is given by $\max_{p(x)} I(U;Y)$, where $X, U$ and $Y$ denote the individual random elements of encoder codewords, decoder codewords and decoder inputs. The capacity result is derived in the "binary symmetric" case (which is an analogous formulation of the traditional "binary symmetric channel" for our case), as a function of the system parameters. A conceptually insightful inference is made by attributing the difference from the classical Shannon-type capacity of binary symmetric channel to the {\em gap} due to the codebook asymmetry.