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
\gamma \beta
-arrays, may be said to be two-dimensional analogs of
M
-sequences. A
\gamma \beta
-array of area
N_x \times N_y
exists over
GF(q)
if and only if
N_x N_y
is equal to
q^N _ 1
for some positive integer
N
. Many interesting characteristics of the
\gamma \beta
-array, such as the properties of its autocorrelation function and the properties of the characteristic arrays, are deduced and explained.