Generalized perfect arrays and menon difference sets

Abstract

Given an s I × ... x s r integer-valued array A and a (0, 1) vector z = (zl, • -., Zr), form the array A' from A by recursively adjoining a negative copy of the current array for each dimension i where zi = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A' are zero, except for shifts (u 1 ..... Ur) where u i =-0 (mod si) for all i. The array is perfect ifz = (0, .... 0) and binary if the array elements are all +1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 22Y32u elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2 y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.

A Unifying Construction for Difference Sets

“…An alternative viewpoint for considering difference sets and RDSs, predominant in engineering papers, is via the correlation properties of binary arrays [27]. The (1, 0) binary array A corresponding to a subset…”

Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

“…In view of this theorem and the remarks at the start of the section, a GPBA of non-zero type can exist only when | G |= 2 or 4t for some t (see also [5, Theorem 8.1(i)]). Further, a PBA can only exist when | G |= 4t 2 for some t (see [5,Theorem 3.1]). Also note that D in the previous theorem is defined in terms of a given a : G → A but that this is not necessary.…”

“…The groups described here were used by Jedwab (see [5]) to connect generalized perfect binary arrays and relative difference sets. We will examine this connection in Section 4.…”

Equivalence classes of central semiregular relative difference sets

“…Generalised perfect binary arrays were introduced by Jedwab [15] for use in signal processing applications requiring sequences with good periodic autocorrelation properties. They include the perfect binary arrays, themselves introduced to overcome the apparent nonexistence of more than a ®nite number of Barker sequences.…”