In the fields of information theory and network coding, a complete characterization of the almost entropic region is a fundamental but inherently difficult open problem. This paper focuses on the characterization of this region via explicit inner bounds, optimization of functions over this region, and network code construction. One approach to study the entropic region is to put a constraint on the alphabets for involved random variables, enabling us to focus on a specific set of joint distributions. We present the notion of alphabet constrained entropic set and describe its properties, namely, its a closed and path-connected set. Motivated by the properties, a random local search algorithm is designed to find an entropic vector and associated distribution near to a given target vector and extended to optimize functions of joint entropies. We present significantly improved inner bound compared to known inner bounds for the almost entropic region involving four random variables using a refinement of our algorithm. Finally, another application of the algorithm is shown to construct network codes and index codes for a given rate vector in the capacity region. This approach to constructing network codes is novel and has an entirely different flavor than classical coding theoretic construction.

This paper focuses on the problem of finding a distribution for an associated entropic vector in the entropy space nearest to a given, possibly non-entropic, target vector for random variables with a constraint on alphabet size. We show the feasibility to find distribution for associated vector via a sequence of perturbations in the probability mass function. Then we present an algorithm for numerically solving the problem together with extensions, applications, and comparison with the known results.