A theory of two-dimensional linear recurring arrays

In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximum-area matrices. We first define two-dimensional linear recurring arrays. In order to study the characteristics of two-dimensional linear recurring arrays, we also define two-dimensional linear cyclic codes. A systematic method of constructing two-dimensional linear recurring arrays having maximum-area matrices is given using the theory of two-dimensional cyclic codes. These arrays, here called -arrays, may be said to be two-dimensional analogs of -sequences. A -array of area exists over if and only if is equal to for some positive integer . Many interesting characteristics of the -array, such as the properties of its autocorrelation function and the properties of the characteristic arrays, are deduced and explained.