Evaluation of the maximum proportion of invertible submatrices in invertible binary matrices

Abstract

The research on all-or-nothing transform in cryptography leads to the problem that, for a given positive integer , what is the density of invertible submatrices of any invertible matrix? For binary matrices, previous work shows that in the case , , where denotes the maximum proportion of invertible submatrices in invertible matrices. In this paper we study the case . It is proved that and an upper bound of is given as well.