An information based criterion for perfectly balanced functions

The class of perfectly balanced functions is important for some areas of mathematics, e. g., combinatorics, coding theory, cryptography, symbolic dynamics, and automata theory. It turns out that perfectly balanced functions provide a suitable mathematical tool for description and studying of convolutional codes, cryptographic primitives, surjective endomorphisms of discrete dynamical systems, and information-lossless finite-state automata. Previously, Hedlund and Sumarokov proved criteria of perfect balancedness of functions, which are related to the property of being defect zero and information-lossless. The present author proves a new criterion of the perfect balancedness property in terms of average mutual information. The author also describes a polinomial-time inverting algorithm for perfectly balanced functions.