On finite sequences satisfying linear recursions

TL;DR

This paper studies sequences satisfying linear recursions over finite fields, deriving formulas for their characteristic functions and analyzing the probability of singular Hankel matrices as sequence length grows.

Contribution

It provides a new algebraic approach to characterize sequences satisfying linear recursions and computes the Fourier transform of their indicator functions over finite fields.

Findings

Characteristic function of H_m expressed as a linear combination of subspace indicators

Derived a formula for the discrete Fourier transform of the characteristic function

Probability of Hankel matrix singularity approaches 1/q under certain conditions

Abstract

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which there exist a_0,...,a_m in k, not all zero, such that sum(a_i x_{i+j}, i=0..m) = 0 holds for each j=0,1,...,n-m. Equivalently, H_m is the set of (x_0,...,x_n) such that the (m+1)-by-(n-m+1) matrix with (i,j) entry x_{i+j} (i=0..m, j=0..n-m) has rank at most m. We use elementary linear and polynomial algebra to study these sets H_m. In particular, when k is a finite field of q elements, we write the characteristic function of H_m as a linear combination of characteristic functions of linear subspaces of dimensions m and m+1 in W_n. We deduce a formula for the discrete Fourier transform (DFT) of this characteristic function, and obtain some consequences. For instance, if the 2m+1 entries of a square Hankel matrix of order m+1 are chosen independently from a fixed but not necessarily uniform distribution mu on k, then as m->infty the matrix is singular with probability approaching 1/q provided the DFT of mu has l_1 norm less than sqrt(q). This bound sqrt(q) is best possible if q is a square.