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 ofM-sequences. A\gamma \beta-array of areaN_x \times N_yexists overGF(q)if and only ifN_x N_yis equal toq^N _ 1for some positive integerN. 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.