Bitwise operations related to a combinatorial problem on binary matrices

TL;DR

This paper explores how bitwise operations can be applied to generate canonical representatives of equivalence classes of binary matrices, addressing an open problem in combinatorial object generation.

Contribution

It introduces a novel algorithm utilizing bitwise operations for isomorphism-free generation of binary matrices with specific row and column properties.

Findings

Algorithm efficiently finds canonical representatives

Bitwise operations enable compact and fast computation

Addresses an open problem in combinatorial object generation

Abstract

Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary matrices having the same number of units in each row and each column is defined. Each binary matrix is represented using ordered n-tuples of natural numbers. It is shown how by using the bitwise operations can be implemented an algorithm that gets canonical representatives which are extremal elements of equivalence classes relative to a double order on the set of considered objects.