Properties of Gröbner Bases and Applications to Doubly Periodic Arrays

Recently, many people have studied two-dimensional arrays over finite fields because they can be used in two-dimensional range-finding, scrambling, two-dimensional cyclic codes and other applications in communication and coding. There are two types of problems on two-dimensional arrays. The first type of problem concerns nonlinear arrays or perfect maps, the second type concerns linear recurring arrays. Nomura et al. (1972) were the first to study linear recurring arrays. In fact they studied linear recurring m-arrays or pseudo-random arrays. In their paper they proposed a problem on the structure of pseudo-random arrays. Siu (1985) proposed that problem again when he visited Beijing. MacWilliams and Sloane (1976) constructed some pseudo-random arrays, Van Lint et al. (1979) studied some linear recurring arrays with special window properties. Lin and Liu solved Nomura’s problem and gave the structure of pseudo-random arrays (Lin and Liu, 1993a; Liu and Li, 1993). Sakata (1978) considered the general theory of linear recurring arrays over finite fields. But further results on the structural general theory of linear recurring arrays were not forthcoming, even though some people (e.g., Lin (1993) and Lin and Liu (1993b)) considered more general linear recurring arrays than pseudo-random arrays, but they were still some specific cases. However, when the Gröbner basis theory was utilized to study linear recurring arrays, some progress was made (Sakata, 1988; Liu and Hu, 1994). In this paper we study doubly periodic arrays, as they are linear recurring arrays and more useful. We apply the properties of Gröbner bases to the space of linear recurring arrays. This yields an explicit basis of the space of linear recurring arrays and gives a pretty, as well as important, trace expression. It is well known that the trace expression of one-dimensional linear recurring sequences is a strong tool for studying their structure and enumeration. Our trace expression of linear recurring arrays can also be used to study their structure as a linear space and as a module, and to calculate the number of translation equivalent classes of linear recurring arrays. But the two-dimensional case is much more complicated than the one-dimensional case. We differ from others in