$S$-constrained random matrices

Sylvain Gravier,Bernard Ycart

doi:10.46298/dmtcs.3480

Abstract

Let $S$ be a set of $d$-dimensional row vectors with entries in a $q$-ary alphabet. A matrix $M$ with entries in the same $q$-ary alphabet is $S$-constrained if every set of $d$ columns of $M$ contains as a submatrix a copy of the vectors in $S$, up to permutation. For a given set $S$ of $d$-dimensional vectors, we compute the asymptotic probability for a random matrix $M$ to be $S$-constrained, as the numbers of rows and columns both tend to infinity. If $n$ is the number of columns and $m=m_n$ the number of rows, then the threshold is at $m_n= \alpha_d \log (n)$, where $\alpha_d$ only depends on the dimension $d$ of vectors and not on the particular set $S$. Applications to superimposed codes, shattering classes of functions, and Sidon families of sets are proposed. For $d=2$, an explicit construction of a $S$-constrained matrix is given.

Highlights

IntroductionWe propose S-constrained matrices as a unifying framework for two seemingly remote notions, one in the field of cryptography (superimposed codes), the other one in information theory (shattering classes of functions)
We propose S-constrained matrices as a unifying framework for two seemingly remote notions, one in the field of cryptography, the other one in information theory
As a consequence of Theorem 1, mn = αd log n is a probabilistic bound for S-constrained matrices: if the number of rows is such that that mn − αd log n tends to −∞, with high probability (w.h.p.), a random matrix is not S-constrained

Summary

Introduction

We propose S-constrained matrices as a unifying framework for two seemingly remote notions, one in the field of cryptography (superimposed codes), the other one in information theory (shattering classes of functions). The columns of the matrix are the words of the code, in the second one, the rows are the functions of the class. 276) that a binary code is called (w, r)-superimposed if for every subsets of words W, R (sets of columns of the matrix) with respective cardinalities w, r, there exists a position (row index) on which every word of W is 1 and every word of R is 0. Let M be q-ary random matrix with mn rows and n columns. As a consequence of Theorem 1, mn = αd log n is a probabilistic bound for S-constrained matrices: if the number of rows is such that that mn − αd log n tends to −∞, with high probability (w.h.p.), a random matrix is not S-constrained.

Proof of Theorem 1

Sidon families

Explicit construction

Concluding remarks