Reflective Numeration Systems I: a Global Standpoint
Provides a theoretical generalization of Gray codes for combinatorial generation, but remains domain-specific to coding theory and combinatorics on words.
The paper introduces a framework to generalize b-ary Gray codes to k-bonacci and other codes using a Z-Gray product, enabling construction of word lists that avoid specified factors and satisfy power-associativity and flipping digit properties.
We present a framework to generalize the standard b-ary Gray code to get the k-bonacci ones obtained in [5] as well as many others by using theoretical tools that allow to make calculations on lists. We introduce the notion of Z-Gray product, from which we deduce sequences of lists of finite words avoiding a predefinite list Z of factors and which satisfy a power-associativity property as well a generalizations of the classical flipping digit property.