On sequences of Toeplitz matrices over finite fields

For each non-negative integer n let An be an n+1 by n+1 Toeplitz matrix over a finite field, F, and suppose for each n that An is embedded in the upper left corner of An+1. We study the structure of the sequence ν={νn:n∈Z+}, where νn=null(An) is the nullity of An. For each n∈Z+ and each nullity pattern ν0,ν1,…,νn, we count the number of strings of Toeplitz matrices A0,A1,…,An with this pattern. As an application we present an elementary proof of a result of D. E. Daykin on the number of n×n Toeplitz matrices over GF(2) of any specified rank.