Perfect maps

Given positive integers r, s, u and v, an (r, s; u, v) Perfect Map (PM) is defined to be a periodic r x s binary array in which every u x v binary array appears exactly once as a subarray. Perfect Maps are the natural extention of the de Bruijin sequences to two dimensions. In this paper we settle the existence question for Perfect Maps by proving the following result. Let r, s, u, v be positive integers. Then there exists an (r, s; u, v) PM if and only if the following three conditions hold: i) rs = 2/sup uv/, ii) r > u or r = u = 1, iii) s > v or s = u = 1. We make extensive use of previously known constructions by finding new conditions guaranteeing their repeated application. These conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of Perfect Maps.