On the existence of square dot-matrix patterns having a specific three-valued periodic-correlation function

Abstract

The author examines square matrices of size n containing dot patterns satisfying the following two restrictions: (1) each column contain precisely one dot, and (2) if the pattern is moved around over a plane tied by the same pattern, when in all positions except the home position there is at most one overlap in dots. From differing viewpoints, there matrices are the characteristic functions of either a certain class of relative difference sets or else a select subset of bent functions. Also, the existence of such an (n*n) matrix implies the existence of a finite projective plane of order n. A family of constructions for such matrices is available when n is prime. A polynomial equation characterizing such matrices and resembling the Hall polynomial equation of cyclic difference sets is presented. Analogs of known existence tests for cyclic difference sets are then applied to rule out existence for most nonprime values of n. It is shown how such patterns can be used to provide hopping patterns for a frequency-hopped multiple-access system.