Variable dimension non-overlapping matrices

TL;DR

This paper introduces non-overlapping sets of binary matrices with variable dimensions by extending string-based non-overlapping concepts to two-dimensional matrices, using pattern avoidance in rows.

Contribution

It extends the concept of non-overlapping strings to variable-dimension matrices, proposing a new method based on pattern avoidance in rows.

Findings

Defined non-overlapping sets of variable-dimension matrices

Constructed matrices avoiding specific consecutive patterns

Extended one-dimensional non-overlapping concepts to two dimensions

Abstract

Since some years, non-overlapping sets of strings (also called cross-bifix-free sets) have had an increasing interest in the frame of the researches about Theory of Codes. Recently some non-overlapping sets of strings with variable length were introduced. Moreover, the notion of non-overlapping strings has been naturally extended to the two dimensional case leading to several definitions of non-overlapping sets of matrices (or pictures). Starting from these results, in this paper we introduce non-overlapping sets of binary matrices having variable dimension. Indeed, we use non-overlapping variable length strings as rows of the matrices and imposing the avoidance of two consecutive patterns of length k, we get the desired sets of non-ovelapping matrices with variable dimension.