Cross-bifix-free sets in two dimensions

TL;DR

This paper extends the concept of bifix-free words to two-dimensional matrices, exhaustively generates bibifix-free matrices, constructs a non-expandable set, and provides a Gray code for listing it.

Contribution

It introduces bidimensional bifix concepts, exhaustively generates bibifix-free matrices, and constructs a non-expandable set with an associated Gray code.

Findings

Exhaustive generation of bibifix-free square matrices

Construction of a non-expandable cross-bibifix-free set

Development of a Gray code for listing the set

Abstract

A bidimensional bifix (in short bibifix) of a square matrix T is a square submatrix of T which occurs in the top-left and bottom-right corners of T. This allows us to extend the definition of bifix-free words and cross-bifix-free set of words to bidimensional structures. In this paper we exhaustively generate all the bibifix-free square matrices and we construct a particular non-expandable cross-bibifix-free set of square matrices. Moreover, we provide a Gray code for listing this set.